Theorem cbvral2vw 2740

Description: Change bound variables of double restricted universal quantification, using implicit substitution. Version of cbvral2v 2742 with a disjoint variable condition, which does not require ax-13 2169. (Contributed by NM, 10-Aug-2004.) (Revised by GG, 10-Jan-2024.)

Hypotheses

Ref	Expression
cbvral2vw.1	|- ( x = z -> ( ph <-> ch ) )
cbvral2vw.2	|- ( y = w -> ( ch <-> ps ) )

Assertion

Ref	Expression
cbvral2vw	|- ( A. x e. A A. y e. B ph <-> A. z e. A A. w e. B ps )

Distinct variable groups:   x, z   y, w   x, A, z   x, y, B, z   w, B   ph, z   ps, y   ch, x   ch, w

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

Proof of Theorem cbvral2vw

Step	Hyp	Ref	Expression
1		cbvral2vw.1	. . . 4 |- ( x = z -> ( ph <-> ch ) )
2	1	ralbidv 2497	. . 3 |- ( x = z -> ( A. y e. B ph <-> A. y e. B ch ) )
3	2	cbvralvw 2733	. 2 |- ( A. x e. A A. y e. B ph <-> A. z e. A A. y e. B ch )
4		cbvral2vw.2	. . . 4 |- ( y = w -> ( ch <-> ps ) )
5	4	cbvralvw 2733	. . 3 |- ( A. y e. B ch <-> A. w e. B ps )
6	5	ralbii 2503	. 2 |- ( A. z e. A A. y e. B ch <-> A. z e. A A. w e. B ps )
7	3, 6	bitri 184	1 |- ( A. x e. A A. y e. B ph <-> A. z e. A A. w e. B ps )

Syntax hints:   -> wi 4   <-> wb 105   A.wral 2475

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-clel 2192  df-ral 2480

This theorem is referenced by:  mhmpropd  13098