Theorem cbvrex2v 2719

Description: Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by FL, 2-Jul-2012.)

Hypotheses

Ref	Expression
cbvrex2v.1	|- ( x = z -> ( ph <-> ch ) )
cbvrex2v.2	|- ( y = w -> ( ch <-> ps ) )

Assertion

Ref	Expression
cbvrex2v	|- ( 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   w, B   x, B, y   z, B, y   ch, w   ch, x   ph, z   ps, y

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

Proof of Theorem cbvrex2v

Step	Hyp	Ref	Expression
1		cbvrex2v.1	. . . 4 |- ( x = z -> ( ph <-> ch ) )
2	1	rexbidv 2478	. . 3 |- ( x = z -> ( E. y e. B ph <-> E. y e. B ch ) )
3	2	cbvrexv 2706	. 2 |- ( E. x e. A E. y e. B ph <-> E. z e. A E. y e. B ch )
4		cbvrex2v.2	. . . 4 |- ( y = w -> ( ch <-> ps ) )
5	4	cbvrexv 2706	. . 3 |- ( E. y e. B ch <-> E. w e. B ps )
6	5	rexbii 2484	. 2 |- ( E. z e. A E. y e. B ch <-> E. z e. A E. w e. B ps )
7	3, 6	bitri 184	1 |- ( E. x e. A E. y e. B ph <-> E. z e. A E. w e. B ps )

Syntax hints:   -> wi 4   <-> wb 105   E.wrex 2456

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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461

This theorem is referenced by:  eroveu  6628  genipv  7510  bezoutlemnewy  11999  xmettx  14095