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Theorem rabbida 3480
Description: Equivalent wff's yield equal restricted class abstractions (deduction form). Version of rabbidva 3484 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.)
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 . . 3 𝑥𝜑
2 rabbida.1 . . . 4 ((𝜑𝑥𝐴) → (𝜓𝜒))
32ex 413 . . 3 (𝜑 → (𝑥𝐴 → (𝜓𝜒)))
41, 3ralrimi 3221 . 2 (𝜑 → ∀𝑥𝐴 (𝜓𝜒))
5 rabbi 3389 . 2 (∀𝑥𝐴 (𝜓𝜒) ↔ {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
64, 5sylib 219 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wnf 1777  wcel 2107  wral 3143  {crab 3147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-ral 3148  df-rab 3152
This theorem is referenced by:  rabbid  3481  bj-rabeqbida  34127  pimgtmnf  42869  smfpimltmpt  42892  smfpimltxrmpt  42904  smfpimgtmpt  42926  smfpimgtxrmpt  42929  smfrec  42933  smfsupmpt  42958  smfinflem  42960  smfinfmpt  42962
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