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Theorem rabeqd 39797
Description: Equality theorem for restricted class abstractions. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypothesis
Ref Expression
rabeqd.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
rabeqd (𝜑 → {𝑥𝐴𝜒} = {𝑥𝐵𝜒})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝜒(𝑥)

Proof of Theorem rabeqd
StepHypRef Expression
1 rabeqd.1 . 2 (𝜑𝐴 = 𝐵)
2 rabeq 3342 . 2 (𝐴 = 𝐵 → {𝑥𝐴𝜒} = {𝑥𝐵𝜒})
31, 2syl 17 1 (𝜑 → {𝑥𝐴𝜒} = {𝑥𝐵𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  {crab 3065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070
This theorem is referenced by:  issmflem  41456  issmfd  41464  cnfsmf  41469  issmflelem  41473  issmfgtlem  41484  issmfgt  41485  issmfled  41486  issmfgtd  41489  issmfgelem  41497
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