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Theorem rabeq 2730
Description: Equality theorem for restricted class abstractions. (Contributed by NM, 15-Oct-2003.)
Assertion
Ref Expression
rabeq (𝐴 = 𝐵 → {𝑥𝐴𝜑} = {𝑥𝐵𝜑})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rabeq
StepHypRef Expression
1 nfcv 2319 . 2 𝑥𝐴
2 nfcv 2319 . 2 𝑥𝐵
31, 2rabeqf 2728 1 (𝐴 = 𝐵 → {𝑥𝐴𝜑} = {𝑥𝐵𝜑})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  {crab 2459
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464
This theorem is referenced by:  rabeqdv  2732  rabeqbidv  2733  rabeqbidva  2734  difeq1  3247  ifeq1  3538  ifeq2  3539  elfvmptrab  5612  pmvalg  6659  unfiexmid  6917  ssfirab  6933  supeq2  6988  iooval2  9915  fzval2  10011  lcmval  12063  lcmcllem  12067  lcmledvds  12070  clsfval  13604
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