Theorem cbvral 2776

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 2386	. 2 |- F/_ x A
2		nfcv 2386	. 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 2771	1 |- ( A. x e. A ph <-> A. y e. A ps )

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1509   A.wral 2522

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527

This theorem is referenced by:  cbvralv  2780  cbvralsv  2796  cbviin  4034  frind  4478  ralxpf  4906  eqfnfv2f  5784  ralrnmpt  5824  dff13f  5949  ofrfval2  6292  uchoice  6344  fmpox  6409  cbvixp  6963  mptelixpg  6982  xpf1o  7110  indstr  9943  fsum3  12098  fsum00  12173  mertenslem2  12247  fprodcl2lem  12316  fprodle  12351  ctiunctal  13276  cnmpt11  15274  cnmpt21  15282  bj-bdfindes  16845  bj-findes  16877