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Theorem rabbida 3459
Description: Equivalent wff's yield equal restricted class abstractions (deduction form). Version of rabbidva 3440 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.) Avoid ax-10 2138, ax-11 2155. (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 580 . 2 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
41, 3rabbida4 3458 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wnf 1786  wcel 2107  {crab 3433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2117  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-rab 3434
This theorem is referenced by:  rabbid  3460  rabeqbida  3462  smfpimltmpt  45462  smfpimltxrmptf  45474  smfpimgtmpt  45497  smfpimgtxrmptf  45500  smfrec  45505  smfsupmpt  45531  smfinflem  45533  smfinfmpt  45535
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