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Theorem frege65c 37242
Description: A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2551 when the minor premise has a general context. Proposition 65 of [Frege1879] p. 53. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
frege59c.a 𝐴𝐵
Assertion
Ref Expression
frege65c (∀𝑥(𝜑𝜓) → (∀𝑥(𝜓𝜒) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜒)))

Proof of Theorem frege65c
StepHypRef Expression
1 sbcim1 3449 . . 3 ([𝐴 / 𝑥](𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))
2 frege59c.a . . . 4 𝐴𝐵
32frege64c 37241 . . 3 (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) → (∀𝑥(𝜓𝜒) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜒)))
41, 3syl 17 . 2 ([𝐴 / 𝑥](𝜑𝜓) → (∀𝑥(𝜓𝜒) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜒)))
52frege61c 37238 . 2 (([𝐴 / 𝑥](𝜑𝜓) → (∀𝑥(𝜓𝜒) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜒))) → (∀𝑥(𝜑𝜓) → (∀𝑥(𝜓𝜒) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜒))))
64, 5ax-mp 5 1 (∀𝑥(𝜑𝜓) → (∀𝑥(𝜓𝜒) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1473  wcel 1977  [wsbc 3402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234  ax-ext 2590  ax-frege1 37104  ax-frege2 37105  ax-frege8 37123  ax-frege58b 37215
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175  df-sbc 3403
This theorem is referenced by:  frege66c  37243
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