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Metamath Proof Explorer

Theorem cbvrex

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbvrexw when possible. (Contributed by NM, 31-Jul-2003) (Proof shortened by Andrew Salmon, 8-Jun-2011) (New usage is discouraged.)

Ref Expression
Hypotheses cbvral.1 
|- F/ y ph
cbvral.2 
|- F/ x ps
cbvral.3 
|- ( x = y -> ( ph <-> ps ) )
Assertion cbvrex 
|- ( E. x e. A ph <-> E. y e. A ps )

Proof

Step Hyp Ref Expression
1 cbvral.1 
 |-  F/ y ph
2 cbvral.2 
 |-  F/ x ps
3 cbvral.3 
 |-  ( x = y -> ( ph <-> ps ) )
4 nfcv 
 |-  F/_ x A
5 nfcv 
 |-  F/_ y A
6 4 5 1 2 3 cbvrexf 
 |-  ( E. x e. A ph <-> E. y e. A ps )