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Theorem rabeq 2714
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 2306 . 2 𝑥𝐴
2 nfcv 2306 . 2 𝑥𝐵
31, 2rabeqf 2712 1 (𝐴 = 𝐵 → {𝑥𝐴𝜑} = {𝑥𝐵𝜑})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1342  {crab 2446
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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-rab 2451
This theorem is referenced by:  rabeqdv  2716  rabeqbidv  2717  rabeqbidva  2718  difeq1  3229  ifeq1  3519  ifeq2  3520  elfvmptrab  5576  pmvalg  6617  unfiexmid  6875  ssfirab  6891  supeq2  6946  iooval2  9843  fzval2  9939  lcmval  11981  lcmcllem  11985  lcmledvds  11988  clsfval  12659
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