Theorem cbvrex2v 3180

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 3052	. . 3 |- ( x = z -> ( E. y e. B ph <-> E. y e. B ch ) )
3	2	cbvrexv 3172	. 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 3172	. . 3 |- ( E. y e. B ch <-> E. w e. B ps )
6	5	rexbii 3041	. 2 |- ( E. z e. A E. y e. B ch <-> E. z e. A E. w e. B ps )
7	3, 6	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   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:  omeu  7665  oeeui  7682  eroveu  7842  genpv  9821  bezoutlem3  15258  bezoutlem4  15259  bezout  15260  4sqlem2  15653  vdwnn  15702  efgrelexlema  18162  dyadmax  23366  2sqlem9  25152  2sq  25155  legov  25480  dfcgra2  25721  pstmfval  29939  nn0prpwlem  32317  isbnd2  33582  fourierdlem42  40366  fourierdlem54  40377  mogoldbb  41673