diff --git a/xarray/core/duck_array_ops.py b/xarray/core/duck_array_ops.py
index 7c0b730814e..9432ccf1904 100644
--- a/xarray/core/duck_array_ops.py
+++ b/xarray/core/duck_array_ops.py
@@ -174,8 +174,10 @@ def as_shared_dtype(scalars_or_arrays):
     return [x.astype(out_type, copy=False) for x in arrays]
 
 
-def allclose_or_equiv(arr1, arr2, rtol=1e-5, atol=1e-8):
-    """Like np.allclose, but also allows values to be NaN in both arrays
+def lazy_array_equiv(arr1, arr2):
+    """Like array_equal, but doesn't actually compare values.
+       Returns True or False when equality can be determined without computing.
+       Returns None when equality cannot determined (e.g. one or both of arr1, arr2 are numpy arrays)
     """
     arr1 = asarray(arr1)
     arr2 = asarray(arr2)
@@ -189,26 +191,19 @@ def allclose_or_equiv(arr1, arr2, rtol=1e-5, atol=1e-8):
         # GH3068
         if arr1.name == arr2.name:
             return True
-    return bool(isclose(arr1, arr2, rtol=rtol, atol=atol, equal_nan=True).all())
+    return None
 
 
-def lazy_array_equiv(arr1, arr2):
-    """Like array_equal, but doesn't actually compare values.
-       Returns True or False when equality can be determined without computing.
-       Returns None when equality cannot determined (e.g. one or both of arr1, arr2 are numpy arrays)
+def allclose_or_equiv(arr1, arr2, rtol=1e-5, atol=1e-8):
+    """Like np.allclose, but also allows values to be NaN in both arrays
     """
     arr1 = asarray(arr1)
     arr2 = asarray(arr2)
-    if arr1.shape != arr2.shape:
-        return False
-    if (
-        dask_array
-        and isinstance(arr1, dask_array.Array)
-        and isinstance(arr2, dask_array.Array)
-    ):
-        # GH3068
-        if arr1.name == arr2.name:
-            return True
+    lazy_equiv = lazy_array_equiv(arr1, arr2)
+    if lazy_equiv is None:
+        return bool(isclose(arr1, arr2, rtol=rtol, atol=atol, equal_nan=True).all())
+    else:
+        return lazy_equiv
 
 
 def array_equiv(arr1, arr2):
@@ -216,20 +211,14 @@ def array_equiv(arr1, arr2):
     """
     arr1 = asarray(arr1)
     arr2 = asarray(arr2)
-    if arr1.shape != arr2.shape:
-        return False
-    if (
-        dask_array
-        and isinstance(arr1, dask_array.Array)
-        and isinstance(arr2, dask_array.Array)
-    ):
-        # GH3068
-        if arr1.name == arr2.name:
-            return True
-    with warnings.catch_warnings():
-        warnings.filterwarnings("ignore", "In the future, 'NAT == x'")
-        flag_array = (arr1 == arr2) | (isnull(arr1) & isnull(arr2))
-        return bool(flag_array.all())
+    lazy_equiv = lazy_array_equiv(arr1, arr2)
+    if lazy_equiv is None:
+        with warnings.catch_warnings():
+            warnings.filterwarnings("ignore", "In the future, 'NAT == x'")
+            flag_array = (arr1 == arr2) | (isnull(arr1) & isnull(arr2))
+            return bool(flag_array.all())
+    else:
+        return lazy_equiv
 
 
 def array_notnull_equiv(arr1, arr2):
@@ -238,20 +227,14 @@ def array_notnull_equiv(arr1, arr2):
     """
     arr1 = asarray(arr1)
     arr2 = asarray(arr2)
-    if arr1.shape != arr2.shape:
-        return False
-    if (
-        dask_array
-        and isinstance(arr1, dask_array.Array)
-        and isinstance(arr2, dask_array.Array)
-    ):
-        # GH3068
-        if arr1.name == arr2.name:
-            return True
-    with warnings.catch_warnings():
-        warnings.filterwarnings("ignore", "In the future, 'NAT == x'")
-        flag_array = (arr1 == arr2) | isnull(arr1) | isnull(arr2)
-        return bool(flag_array.all())
+    lazy_equiv = lazy_array_equiv(arr1, arr2)
+    if lazy_equiv is None:
+        with warnings.catch_warnings():
+            warnings.filterwarnings("ignore", "In the future, 'NAT == x'")
+            flag_array = (arr1 == arr2) | isnull(arr1) | isnull(arr2)
+            return bool(flag_array.all())
+    else:
+        return lazy_equiv
 
 
 def count(data, axis=None):