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Theorem rabbida 39795
Description: Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
rabbida.1 𝑥𝜑
rabbida.2 ((𝜑𝑥𝐴) → (𝜓𝜒))
Assertion
Ref Expression
rabbida (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})

Proof of Theorem rabbida
StepHypRef Expression
1 rabbida.1 . . 3 𝑥𝜑
2 rabbida.2 . . . 4 ((𝜑𝑥𝐴) → (𝜓𝜒))
32ex 397 . . 3 (𝜑 → (𝑥𝐴 → (𝜓𝜒)))
41, 3ralrimi 3106 . 2 (𝜑 → ∀𝑥𝐴 (𝜓𝜒))
5 rabbi 3269 . 2 (∀𝑥𝐴 (𝜓𝜒) ↔ {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
64, 5sylib 208 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wnf 1856  wcel 2145  wral 3061  {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-ral 3066  df-rab 3070
This theorem is referenced by:  pimgtmnf  41452  smfpimltmpt  41475  smfpimltxrmpt  41487  smfpimgtmpt  41509  smfpimgtxrmpt  41512  smfrec  41516  smfsupmpt  41541  smfinflem  41543  smfinfmpt  41545
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