Theorem cbvral 2700

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 2319	. 2 |- F/_ x A
2		nfcv 2319	. 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 2697	1 |- ( A. x e. A ph <-> A. y e. A ps )

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1460   A.wral 2455

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-nf 1461  df-sb 1763  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460

This theorem is referenced by:  cbvralv  2704  cbvralsv  2720  cbviin  3925  frind  4353  ralxpf  4774  eqfnfv2f  5618  ralrnmpt  5659  dff13f  5771  ofrfval2  6099  fmpox  6201  cbvixp  6715  mptelixpg  6734  xpf1o  6844  indstr  9593  fsum3  11395  fsum00  11470  mertenslem2  11544  fprodcl2lem  11613  fprodle  11648  ctiunctal  12442  cnmpt11  13786  cnmpt21  13794  bj-bdfindes  14704  bj-findes  14736