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Theorem rabbida 3456
Description: Equivalent wff's yield equal restricted class abstractions (deduction form). Version of rabbidva 3437 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.) Avoid ax-10 2135, ax-11 2152. (Revised by Wolf Lammen, 14-Mar-2025.)
Hypotheses
Ref Expression
rabbida.n 𝑥𝜑
rabbida.1 ((𝜑𝑥𝐴) → (𝜓𝜒))
Assertion
Ref Expression
rabbida (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})

Proof of Theorem rabbida
StepHypRef Expression
1 rabbida.n . 2 𝑥𝜑
2 rabbida.1 . . 3 ((𝜑𝑥𝐴) → (𝜓𝜒))
32pm5.32da 577 . 2 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
41, 3rabbida4 3455 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1539  wnf 1783  wcel 2104  {crab 3430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-9 2114  ax-12 2169  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-rab 3431
This theorem is referenced by:  rabbid  3457  rabeqbida  3459  smfpimltmpt  45760  smfpimltxrmptf  45772  smfpimgtmpt  45795  smfpimgtxrmptf  45798  smfrec  45803  smfsupmpt  45829  smfinflem  45831  smfinfmpt  45833
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