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Theorem rabbida 3461
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 2139, 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 579 . 2 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
41, 3rabbida4 3460 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wnf 1780  wcel 2106  {crab 3433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-9 2116  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-rab 3434
This theorem is referenced by:  rabbid  3462  rabeqbida  3464  smfpimltmpt  46702  smfpimltxrmptf  46714  smfpimgtmpt  46737  smfpimgtxrmptf  46740  smfrec  46745  smfsupmpt  46771  smfinflem  46773  smfinfmpt  46775
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