Theorem cbvral 2735

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 2349	. 2 |- F/_ x A
2		nfcv 2349	. 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 2731	1 |- ( A. x e. A ph <-> A. y e. A ps )

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1484   A.wral 2485

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 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490

This theorem is referenced by:  cbvralv  2739  cbvralsv  2755  cbviin  3974  frind  4412  ralxpf  4837  eqfnfv2f  5699  ralrnmpt  5740  dff13f  5857  ofrfval2  6193  uchoice  6241  fmpox  6304  cbvixp  6820  mptelixpg  6839  xpf1o  6961  indstr  9744  fsum3  11783  fsum00  11858  mertenslem2  11932  fprodcl2lem  12001  fprodle  12036  ctiunctal  12897  cnmpt11  14840  cnmpt21  14848  bj-bdfindes  16054  bj-findes  16086