Theorem cbvral 2738

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 2350	. 2 |- F/_ x A
2		nfcv 2350	. 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 2733	1 |- ( A. x e. A ph <-> A. y e. A ps )

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1484   A.wral 2486

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491

This theorem is referenced by:  cbvralv  2742  cbvralsv  2758  cbviin  3979  frind  4417  ralxpf  4842  eqfnfv2f  5704  ralrnmpt  5745  dff13f  5862  ofrfval2  6198  uchoice  6246  fmpox  6309  cbvixp  6825  mptelixpg  6844  xpf1o  6966  indstr  9749  fsum3  11813  fsum00  11888  mertenslem2  11962  fprodcl2lem  12031  fprodle  12066  ctiunctal  12927  cnmpt11  14870  cnmpt21  14878  bj-bdfindes  16084  bj-findes  16116