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  6005  exmidac  7402  genipv  7707  ltexpri  7811  suplocsrlempr  8005  suplocsr  8007  zsupssdc  10470  bitsfzolem  12481  nninfctlemfo  12577  sqne2sq  12715  eulerth  12771  odzval  12780  pcprecl  12828  pcprendvds  12829  pcpremul  12832  pceulem  12833  4sqlem19  12948  lfgredg2dom  15946  vtxdumgrfival  16058  vtxduspgrfvedgfilem  16060  vtxduspgrfvedgfi  16061