Theorem cbvrabv 2600

Description: Rule to change the bound variable in a restricted class abstraction, using implicit substitution. (Contributed by NM, 26-May-1999.)

Hypothesis

Ref	Expression
cbvrabv.1	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvrabv	|- { x e. A | ph } = { y e. A | ps }

Distinct variable groups:   x, y, A   ph, y   ps, x

Allowed substitution hints:   ph( x)   ps( y)

Proof of Theorem cbvrabv

Step	Hyp	Ref	Expression
1		nfcv 2219	. 2 |- F/_ x A
2		nfcv 2219	. 2 |- F/_ y A
3		nfv 1461	. 2 |- F/ y ph
4		nfv 1461	. 2 |- F/ x ps
5		cbvrabv.1	. 2 |- ( x = y -> ( ph <-> ps ) )
6	1, 2, 3, 4, 5	cbvrab 2599	1 |- { x e. A | ph } = { y e. A | ps }

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   {crab 2352

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rab 2357

This theorem is referenced by:  pwnss  3933  acexmidlemv  5530  genipv  6699  ltexpri  6803  sqne2sq  10555