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Theorem rabbidaOLD 3467
Description: Obsolete version of rabbida 3455 as of 14-Mar-2025. (Contributed by BJ, 27-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
rabbidaOLD.n 𝑥𝜑
rabbidaOLD.1 ((𝜑𝑥𝐴) → (𝜓𝜒))
Assertion
Ref Expression
rabbidaOLD (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})

Proof of Theorem rabbidaOLD
StepHypRef Expression
1 rabbidaOLD.n . . 3 𝑥𝜑
2 rabbidaOLD.1 . . . 4 ((𝜑𝑥𝐴) → (𝜓𝜒))
32ex 412 . . 3 (𝜑 → (𝑥𝐴 → (𝜓𝜒)))
41, 3ralrimi 3251 . 2 (𝜑 → ∀𝑥𝐴 (𝜓𝜒))
5 rabbi 3459 . 2 (∀𝑥𝐴 (𝜓𝜒) ↔ {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
64, 5sylib 217 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  wnf 1778  wcel 2099  wral 3058  {crab 3429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-ral 3059  df-rab 3430
This theorem is referenced by: (None)
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