Theorem rabeq 2768

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 2350	. 2 |- F/_ x A
2		nfcv 2350	. 2 |- F/_ x B
3	1, 2	rabeqf 2766	1 |- ( A = B -> { x e. A | ph } = { x e. B | ph } )

Syntax hints:   -> wi 4   = wceq 1373   {crab 2490

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rab 2495

This theorem is referenced by:  rabeqdv  2770  rabeqbidv  2771  rabeqbidva  2772  difeq1  3292  ifeq1  3582  ifeq2  3583  elfvmptrab  5698  pmvalg  6769  unfiexmid  7041  ssfirab  7059  supeq2  7117  iooval2  10072  fzval2  10168  clsfval  14688  incistruhgr  15801