Theorem cbvral 2733

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 2347	. 2 |- F/_ x A
2		nfcv 2347	. 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 2729	1 |- ( A. x e. A ph <-> A. y e. A ps )

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1482   A.wral 2483

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488

This theorem is referenced by:  cbvralv  2737  cbvralsv  2753  cbviin  3964  frind  4398  ralxpf  4823  eqfnfv2f  5680  ralrnmpt  5721  dff13f  5838  ofrfval2  6174  uchoice  6222  fmpox  6285  cbvixp  6801  mptelixpg  6820  xpf1o  6940  indstr  9713  fsum3  11669  fsum00  11744  mertenslem2  11818  fprodcl2lem  11887  fprodle  11922  ctiunctal  12783  cnmpt11  14726  cnmpt21  14734  bj-bdfindes  15847  bj-findes  15879