Theorem cbval2vw 1944

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 4-Feb-2005.) (Revised by GG, 10-Jan-2024.)

Hypothesis

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

Assertion

Ref	Expression
cbval2vw	|- ( A. x A. y ph <-> A. z A. w ps )

Distinct variable groups:   z, w, ph   x, y, ps   x, w, y, z

Allowed substitution hints:   ph( x, y)   ps( z, w)

Proof of Theorem cbval2vw

Step	Hyp	Ref	Expression
1		cbval2vw.1	. . 3 |- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )
2	1	cbvaldvaw 1942	. 2 |- ( x = z -> ( A. y ph <-> A. w ps ) )
3	2	cbvalvw 1931	1 |- ( A. x A. y ph <-> A. z A. w ps )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546

This theorem depends on definitions:  df-bi 117  df-nf 1472

This theorem is referenced by:  seqf1og  10592