Theorem cbvrex2 41173

Description: Change bound variables of double restricted universal quantification, using implicit substitution, analogous to cbvrex2v 3180. (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
cbvrex2	|- ( E. x e. A E. y e. B ph <-> E. z e. A E. 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 cbvrex2

Step	Hyp	Ref	Expression
1		nfcv 2764	. . . 4 |- F/_ z B
2		cbvral2.1	. . . 4 |- F/ z ph
3	1, 2	nfrex 3007	. . 3 |- F/ z E. y e. B ph
4		nfcv 2764	. . . 4 |- F/_ x B
5		cbvral2.2	. . . 4 |- F/ x ch
6	4, 5	nfrex 3007	. . 3 |- F/ x E. y e. B ch
7		cbvral2.5	. . . 4 |- ( x = z -> ( ph <-> ch ) )
8	7	rexbidv 3052	. . 3 |- ( x = z -> ( E. y e. B ph <-> E. y e. B ch ) )
9	3, 6, 8	cbvrex 3168	. 2 |- ( E. x e. A E. y e. B ph <-> E. z e. A E. 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	cbvrex 3168	. . 3 |- ( E. y e. B ch <-> E. w e. B ps )
14	13	rexbii 3041	. 2 |- ( E. z e. A E. y e. B ch <-> E. z e. A E. w e. B ps )
15	9, 14	bitri 264	1 |- ( E. x e. A E. y e. B ph <-> E. z e. A E. w e. B ps )

Syntax hints:   -> wi 4   <-> wb 196   F/wnf 1708   E.wrex 2913

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  df-rex 2918

This theorem is referenced by: (None)