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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 
|- ( -. A. x x = y -> F/_ x y )

Proof

Step Hyp Ref Expression
1 nfv 
 |-  F/ w -. A. x x = y
2 nfv 
 |-  F/ x w e. z
3 elequ2 
 |-  ( z = y -> ( w e. z <-> w e. y ) )
4 2 3 dvelimnf 
 |-  ( -. A. x x = y -> F/ x w e. y )
5 1 4 nfcd 
 |-  ( -. A. x x = y -> F/_ x y )