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Metamath Proof Explorer

Theorem cbvreuv

Description: Change the bound variable of a restricted unique existential quantifier using implicit substitution. See cbvreuvw for a version without ax-13 , but extra disjoint variables. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbvreuvw when possible. (Contributed by NM, 5-Apr-2004) (Revised by Mario Carneiro, 15-Oct-2016) (New usage is discouraged.)

Ref Expression
Hypothesis cbvrmov.1 
|- ( x = y -> ( ph <-> ps ) )
Assertion cbvreuv 
|- ( E! x e. A ph <-> E! y e. A ps )

Proof

Step Hyp Ref Expression
1 cbvrmov.1 
 |-  ( x = y -> ( ph <-> ps ) )
2 nfv 
 |-  F/ y ph
3 nfv 
 |-  F/ x ps
4 2 3 1 cbvreu 
 |-  ( E! x e. A ph <-> E! y e. A ps )