Theorem cbvral 2761

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.)

Hypotheses

Ref	Expression
cbvral.1	|- F/ y ph
cbvral.2	|- F/ x ps
cbvral.3	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvral	|- ( A. x e. A ph <-> A. y e. A ps )

Distinct variable groups:   x, A   y, A

Allowed substitution hints:   ph( x, y)   ps( x, y)

Proof of Theorem cbvral

Step	Hyp	Ref	Expression
1		nfcv 2372	. 2 |- F/_ x A
2		nfcv 2372	. 2 |- F/_ y A
3		cbvral.1	. 2 |- F/ y ph
4		cbvral.2	. 2 |- F/ x ps
5		cbvral.3	. 2 |- ( x = y -> ( ph <-> ps ) )
6	1, 2, 3, 4, 5	cbvralf 2756	1 |- ( A. x e. A ph <-> A. y e. A ps )

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1506   A.wral 2508

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513

This theorem is referenced by:  cbvralv  2765  cbvralsv  2781  cbviin  4006  frind  4447  ralxpf  4874  eqfnfv2f  5744  ralrnmpt  5785  dff13f  5906  ofrfval2  6247  uchoice  6295  fmpox  6360  cbvixp  6879  mptelixpg  6898  xpf1o  7025  indstr  9817  fsum3  11938  fsum00  12013  mertenslem2  12087  fprodcl2lem  12156  fprodle  12191  ctiunctal  13052  cnmpt11  14997  cnmpt21  15005  bj-bdfindes  16480  bj-findes  16512