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Theorem rabeqd 38798
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 3183 . 2 (𝐴 = 𝐵 → {𝑥𝐴𝜒} = {𝑥𝐵𝜒})
31, 2syl 17 1 (𝜑 → {𝑥𝐴𝜒} = {𝑥𝐵𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  {crab 2912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2917
This theorem is referenced by:  issmflem  40273  issmfd  40281  cnfsmf  40286  issmflelem  40290  issmfgtlem  40301  issmfgt  40302  issmfled  40303  issmfgtd  40306  issmfgelem  40314
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