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Metamath Proof Explorer

Theorem cbvexv

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . See cbvexvw for a version requiring fewer axioms, to be preferred when sufficient. (Contributed by NM, 21-Jun-1993) Remove dependency on ax-10 and shorten proof. (Revised by Wolf Lammen, 11-Sep-2023) (New usage is discouraged.)

Ref Expression
Hypothesis cbvalv.1 
|- ( x = y -> ( ph <-> ps ) )
Assertion cbvexv 
|- ( E. x ph <-> E. y ps )

Proof

Step Hyp Ref Expression
1 cbvalv.1 
 |-  ( x = y -> ( ph <-> ps ) )
2 nfv 
 |-  F/ y ph
3 nfv 
 |-  F/ x ps
4 2 3 1 cbvex 
 |-  ( E. x ph <-> E. y ps )