Theorem rabeq 2678

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 2281	. 2 |- F/_ x A
2		nfcv 2281	. 2 |- F/_ x B
3	1, 2	rabeqf 2676	1 |- ( A = B -> { x e. A | ph } = { x e. B | ph } )

Syntax hints:   -> wi 4   = wceq 1331   {crab 2420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rab 2425

This theorem is referenced by:  rabeqdv  2680  rabeqbidv  2681  rabeqbidva  2682  difeq1  3187  ifeq1  3477  ifeq2  3478  elfvmptrab  5516  pmvalg  6553  unfiexmid  6806  ssfirab  6822  supeq2  6876  iooval2  9698  fzval2  9793  lcmval  11744  lcmcllem  11748  lcmledvds  11751  clsfval  12270