Theorem rabeq 2805

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 2384	. 2 |- F/_ x A
2		nfcv 2384	. 2 |- F/_ x B
3	1, 2	rabeqf 2803	1 |- ( A = B -> { x e. A | ph } = { x e. B | ph } )

Syntax hints:   -> wi 4   = wceq 1398   {crab 2524

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 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rab 2529

This theorem is referenced by:  rabeqdv  2807  rabeqbidv  2808  rabeqbidva  2809  difeq1  3330  ifeq1  3625  ifeq2  3626  elfvmptrab  5773  supp0  6438  pmvalg  6893  unfiexmid  7178  ssfirab  7197  supeq2  7280  iooval2  10248  fzval2  10345  clsfval  14966  incistruhgr  16085