Theorem cbvral 2734

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 2348	. 2 |- F/_ x A
2		nfcv 2348	. 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 2730	1 |- ( A. x e. A ph <-> A. y e. A ps )

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1483   A.wral 2484

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489

This theorem is referenced by:  cbvralv  2738  cbvralsv  2754  cbviin  3965  frind  4399  ralxpf  4824  eqfnfv2f  5681  ralrnmpt  5722  dff13f  5839  ofrfval2  6175  uchoice  6223  fmpox  6286  cbvixp  6802  mptelixpg  6821  xpf1o  6941  indstr  9714  fsum3  11698  fsum00  11773  mertenslem2  11847  fprodcl2lem  11916  fprodle  11951  ctiunctal  12812  cnmpt11  14755  cnmpt21  14763  bj-bdfindes  15885  bj-findes  15917