Metamath Proof Explorer

Theorem nfcvf

Description: If x and y are distinct, then x is not free in y . Usage of this theorem is discouraged because it depends on ax-13 . See nfcv for a version that replaces the distinctor with a disjoint variable condition, requiring fewer axioms. (Contributed by Mario Carneiro, 8-Oct-2016) Avoid ax-ext . (Revised by Wolf Lammen, 10-May-2023) (New usage is discouraged.)

Ref Expression
Assertion nfcvf ¬ x x = y _ x y


Step Hyp Ref Expression
1 nfv w ¬ x x = y
2 nfv x w z
3 elequ2 z = y w z w y
4 2 3 dvelimnf ¬ x x = y x w y
5 1 4 nfcd ¬ x x = y _ x y