Theorem cbvral2 41172

Description: Change bound variables of double restricted universal quantification, using implicit substitution, analogous to cbvral2v 3179. (Contributed by Alexander van der Vekens, 2-Jul-2017.)

Hypotheses

Ref	Expression
cbvral2.1	|- F/ z ph
cbvral2.2	|- F/ x ch
cbvral2.3	|- F/ w ch
cbvral2.4	|- F/ y ps
cbvral2.5	|- ( x = z -> ( ph <-> ch ) )
cbvral2.6	|- ( y = w -> ( ch <-> ps ) )

Assertion

Ref	Expression
cbvral2	|- ( A. x e. A A. y e. B ph <-> A. z e. A A. w e. B ps )

Distinct variable groups:   x, A   z, A   x, y, B   y, z, B   w, B

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

Proof of Theorem cbvral2

Step	Hyp	Ref	Expression
1		nfcv 2764	. . . 4 |- F/_ z B
2		cbvral2.1	. . . 4 |- F/ z ph
3	1, 2	nfral 2945	. . 3 |- F/ z A. y e. B ph
4		nfcv 2764	. . . 4 |- F/_ x B
5		cbvral2.2	. . . 4 |- F/ x ch
6	4, 5	nfral 2945	. . 3 |- F/ x A. y e. B ch
7		cbvral2.5	. . . 4 |- ( x = z -> ( ph <-> ch ) )
8	7	ralbidv 2986	. . 3 |- ( x = z -> ( A. y e. B ph <-> A. y e. B ch ) )
9	3, 6, 8	cbvral 3167	. 2 |- ( A. x e. A A. y e. B ph <-> A. z e. A A. y e. B ch )
10		cbvral2.3	. . . 4 |- F/ w ch
11		cbvral2.4	. . . 4 |- F/ y ps
12		cbvral2.6	. . . 4 |- ( y = w -> ( ch <-> ps ) )
13	10, 11, 12	cbvral 3167	. . 3 |- ( A. y e. B ch <-> A. w e. B ps )
14	13	ralbii 2980	. 2 |- ( A. z e. A A. y e. B ch <-> A. z e. A A. w e. B ps )
15	9, 14	bitri 264	1 |- ( A. x e. A A. y e. B ph <-> A. z e. A A. w e. B ps )

Syntax hints:   -> wi 4   <-> wb 196   F/wnf 1708   A.wral 2912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917

This theorem is referenced by: (None)