Theorem rabeq 2752

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 2336	. 2 |- F/_ x A
2		nfcv 2336	. 2 |- F/_ x B
3	1, 2	rabeqf 2750	1 |- ( A = B -> { x e. A | ph } = { x e. B | ph } )

Syntax hints:   -> wi 4   = wceq 1364   {crab 2476

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rab 2481

This theorem is referenced by:  rabeqdv  2754  rabeqbidv  2755  rabeqbidva  2756  difeq1  3270  ifeq1  3560  ifeq2  3561  elfvmptrab  5653  pmvalg  6713  unfiexmid  6974  ssfirab  6990  supeq2  7048  iooval2  9981  fzval2  10077  clsfval  14269