Theorem rabeq 2807

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 2386	. 2 |- F/_ x A
2		nfcv 2386	. 2 |- F/_ x B
3	1, 2	rabeqf 2805	1 |- ( A = B -> { x e. A | ph } = { x e. B | ph } )

Syntax hints:   -> wi 4   = wceq 1398   {crab 2526

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rab 2531

This theorem is referenced by:  rabeqdv  2809  rabeqbidv  2810  rabeqbidva  2811  difeq1  3334  ifeq1  3629  ifeq2  3630  elfvmptrab  5778  supp0  6451  pmvalg  6906  unfiexmid  7191  ssfirab  7210  supeq2  7293  iooval2  10267  fzval2  10364  clsfval  15092  incistruhgr  16211