Theorem cbvrabv 2798

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 2372	. 2 |- F/_ x A
2		nfcv 2372	. 2 |- F/_ y A
3		nfv 1574	. 2 |- F/ y ph
4		nfv 1574	. 2 |- F/ x ps
5		cbvrabv.1	. 2 |- ( x = y -> ( ph <-> ps ) )
6	1, 2, 3, 4, 5	cbvrab 2797	1 |- { x e. A | ph } = { y e. A | ps }

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   {crab 2512

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rab 2517

This theorem is referenced by:  pwnss  4243  acexmidlemv  5999  exmidac  7391  genipv  7696  ltexpri  7800  suplocsrlempr  7994  suplocsr  7996  zsupssdc  10458  bitsfzolem  12465  nninfctlemfo  12561  sqne2sq  12699  eulerth  12755  odzval  12764  pcprecl  12812  pcprendvds  12813  pcpremul  12816  pceulem  12817  4sqlem19  12932  lfgredg2dom  15930