Theorem rabeq 2612

Description: Equality theorem for restricted class abstractions. (Contributed by NM, 15-Oct-2003.)

Assertion

Ref	Expression
rabeq	|- ( A = B -> { x e. A | ph } = { x e. B | ph } )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   ph( x)

Proof of Theorem rabeq

Step	Hyp	Ref	Expression
1		nfcv 2229	. 2 |- F/_ x A
2		nfcv 2229	. 2 |- F/_ x B
3	1, 2	rabeqf 2610	1 |- ( A = B -> { x e. A | ph } = { x e. B | ph } )

Syntax hints:   -> wi 4   = wceq 1290   {crab 2364

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rab 2369

This theorem is referenced by:  rabeqdv  2614  rabeqbidv  2615  rabeqbidva  2616  difeq1  3112  ifeq1  3400  ifeq2  3401  pmvalg  6430  unfiexmid  6682  ssfirab  6697  supeq2  6738  iooval2  9394  fzval2  9488  lcmval  11384  lcmcllem  11388  lcmledvds  11391  clsfval  11862