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Theorem rabbida 38302
 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 449 . . 3 (𝜑 → (𝑥𝐴 → (𝜓𝜒)))
41, 3ralrimi 2940 . 2 (𝜑 → ∀𝑥𝐴 (𝜓𝜒))
5 rabbi 3097 . 2 (∀𝑥𝐴 (𝜓𝜒) ↔ {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
64, 5sylib 207 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  ∀wral 2896  {crab 2900 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-ral 2901  df-rab 2905 This theorem is referenced by:  pimgtmnf  39609  smfpimltmpt  39633  smfpimltxrmpt  39645  smfpimgtmpt  39667  smfpimgtxrmpt  39670  smfrec  39674
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