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Theorem rabbida 40033
Description: Equivalent wff's yield equal restricted class abstractions (deduction form). (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 402 . . 3 (𝜑 → (𝑥𝐴 → (𝜓𝜒)))
41, 3ralrimi 3138 . 2 (𝜑 → ∀𝑥𝐴 (𝜓𝜒))
5 rabbi 3302 . 2 (∀𝑥𝐴 (𝜓𝜒) ↔ {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
64, 5sylib 210 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wnf 1879  wcel 2157  wral 3089  {crab 3093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-ral 3094  df-rab 3098
This theorem is referenced by:  pimgtmnf  41678  smfpimltmpt  41701  smfpimltxrmpt  41713  smfpimgtmpt  41735  smfpimgtxrmpt  41738  smfrec  41742  smfsupmpt  41767  smfinflem  41769  smfinfmpt  41771
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