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Theorem rabeqdv 2743
Description: Equality of restricted class abstractions. Deduction form of rabeq 2741. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
Hypothesis
Ref Expression
rabeqdv.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
rabeqdv (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜓})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem rabeqdv
StepHypRef Expression
1 rabeqdv.1 . 2 (𝜑𝐴 = 𝐵)
2 rabeq 2741 . 2 (𝐴 = 𝐵 → {𝑥𝐴𝜓} = {𝑥𝐵𝜓})
31, 2syl 14 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜓})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1363  {crab 2469
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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rab 2474
This theorem is referenced by:  dfphi2  12234  lspfval  13577  lsppropd  13621  cncfval  14355  reldvg  14444  dvfvalap  14446
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