Theorem cbvral2v 3179

Description: Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by NM, 10-Aug-2004.)

Hypotheses

Ref	Expression
cbvral2v.1	|- ( x = z -> ( ph <-> ch ) )
cbvral2v.2	|- ( y = w -> ( ch <-> ps ) )

Assertion

Ref	Expression
cbvral2v	|- ( 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   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 cbvral2v

Step	Hyp	Ref	Expression
1		cbvral2v.1	. . . 4 |- ( x = z -> ( ph <-> ch ) )
2	1	ralbidv 2986	. . 3 |- ( x = z -> ( A. y e. B ph <-> A. y e. B ch ) )
3	2	cbvralv 3171	. 2 |- ( A. x e. A A. y e. B ph <-> A. z e. A A. y e. B ch )
4		cbvral2v.2	. . . 4 |- ( y = w -> ( ch <-> ps ) )
5	4	cbvralv 3171	. . 3 |- ( A. y e. B ch <-> A. w e. B ps )
6	5	ralbii 2980	. 2 |- ( A. z e. A A. y e. B ch <-> A. z e. A A. w e. B ps )
7	3, 6	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   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:  cbvral3v  3181  fununi  5964  fiint  8237  nqereu  9751  mhmpropd  17341  efgred  18161  mplcoe5  19468  mdetunilem9  20426  fbun  21644  fbunfip  21673  caucfil  23081  pmltpc  23219  iscgrglt  25409  axcontlem10  25853  htth  27775  cdj3lem3b  29299  cdj3i  29300  isros  30231  rossros  30243  fipjust  37870  isotone1  38346  isotone2  38347  ntrclsiso  38365  ntrclskb  38367  ntrclsk3  38368  ntrclsk13  38369  pimincfltioo  40928  incsmf  40951  decsmf  40975  mgmhmpropd  41785