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Metamath Proof Explorer

Theorem cbval2vv

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 cbval2vw if possible. (Contributed by NM, 4-Feb-2005) Remove dependency on ax-10 . (Revised by Wolf Lammen, 18-Jul-2021) (New usage is discouraged.)

Ref Expression
Hypothesis cbval2vv.1 
|- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )
Assertion cbval2vv 
|- ( A. x A. y ph <-> A. z A. w ps )

Proof

Step Hyp Ref Expression
1 cbval2vv.1 
 |-  ( ( x = z /\ y = w ) -> ( ph <-> ps ) )
2 1 cbvaldva 
 |-  ( x = z -> ( A. y ph <-> A. w ps ) )
3 2 cbvalv 
 |-  ( A. x A. y ph <-> A. z A. w ps )