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Theorem rabbida 3463
Description: Equivalent wff's yield equal restricted class abstractions (deduction form). Version of rabbidva 3443 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.) Avoid ax-10 2141, ax-11 2157. (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 3462 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wnf 1783  wcel 2108  {crab 3436
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 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-rab 3437
This theorem is referenced by:  rabbid  3464  rabeqbida  3466  smfpimltmpt  46761  smfpimltxrmptf  46773  smfpimgtmpt  46796  smfpimgtxrmptf  46799  smfrec  46804  smfsupmpt  46830  smfinflem  46832  smfinfmpt  46834
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