Theorem rabeq 2813

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 2392	. 2 |- F/_ x A
2		nfcv 2392	. 2 |- F/_ x B
3	1, 2	rabeqf 2811	1 |- ( A = B -> { x e. A | ph } = { x e. B | ph } )

Syntax hints:   -> wi 4   = wceq 1402   {crab 2532

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rab 2537

This theorem is referenced by:  rabeqdv  2815  rabeqbidv  2816  rabeqbidva  2817  difeq1  3340  ifeq1  3640  ifeq2  3641  elfvmptrab  5795  supp0  6468  pmvalg  6923  unfiexmid  7215  ssfirab  7234  supeq2  7319  iooval2  10296  fzval2  10393  clsfval  15125  incistruhgr  16245