Theorem cbvral 2764

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 2375	. 2 |- F/_ x A
2		nfcv 2375	. 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 2759	1 |- ( A. x e. A ph <-> A. y e. A ps )

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1509   A.wral 2511

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 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516

This theorem is referenced by:  cbvralv  2768  cbvralsv  2784  cbviin  4013  frind  4455  ralxpf  4882  eqfnfv2f  5757  ralrnmpt  5797  dff13f  5921  ofrfval2  6261  uchoice  6309  fmpox  6374  cbvixp  6927  mptelixpg  6946  xpf1o  7073  indstr  9871  fsum3  12011  fsum00  12086  mertenslem2  12160  fprodcl2lem  12229  fprodle  12264  ctiunctal  13125  cnmpt11  15077  cnmpt21  15085  bj-bdfindes  16648  bj-findes  16680