Theorem cbvral 2774

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 2384	. 2 |- F/_ x A
2		nfcv 2384	. 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 2769	1 |- ( A. x e. A ph <-> A. y e. A ps )

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1509   A.wral 2520

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 2214

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525

This theorem is referenced by:  cbvralv  2778  cbvralsv  2794  cbviin  4029  frind  4473  ralxpf  4901  eqfnfv2f  5779  ralrnmpt  5819  dff13f  5943  ofrfval2  6283  uchoice  6331  fmpox  6396  cbvixp  6950  mptelixpg  6969  xpf1o  7097  indstr  9925  fsum3  12073  fsum00  12148  mertenslem2  12222  fprodcl2lem  12291  fprodle  12326  ctiunctal  13192  cnmpt11  15148  cnmpt21  15156  bj-bdfindes  16719  bj-findes  16751