Theorem cbvral 2586

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 2228	. 2 |- F/_ x A
2		nfcv 2228	. 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 2584	1 |- ( A. x e. A ph <-> A. y e. A ps )

Syntax hints:   -> wi 4   <-> wb 103   F/wnf 1394   A.wral 2359

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364

This theorem is referenced by:  cbvralv  2590  cbvralsv  2601  cbviin  3768  frind  4179  ralxpf  4582  eqfnfv2f  5401  ralrnmpt  5441  dff13f  5549  ofrfval2  5871  fmpt2x  5970  xpf1o  6558  indstr  9079  fisum  10774  fsum00  10852  mertenslem2  10926  bj-bdfindes  11799  bj-findes  11831