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Metamath Proof Explorer

Theorem cbval2vw

Description: Rule used to change bound variables, using implicit substitution. Version of cbval2vv with more disjoint variable conditions, which requires fewer axioms . (Contributed by NM, 4-Feb-2005) Avoid ax-13 . (Revised by GG, 10-Jan-2024)

Ref Expression
Hypothesis cbval2vw.1 
|- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )
Assertion cbval2vw 
|- ( A. x A. y ph <-> A. z A. w ps )

Proof

Step Hyp Ref Expression
1 cbval2vw.1 
 |-  ( ( x = z /\ y = w ) -> ( ph <-> ps ) )
2 1 cbvaldvaw 
 |-  ( x = z -> ( A. y ph <-> A. w ps ) )
3 2 cbvalvw 
 |-  ( A. x A. y ph <-> A. z A. w ps )