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Theorem frege65c 39063
 Description: A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2747 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 3710 . . 3 ([𝐴 / 𝑥](𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))
2 frege59c.a . . . 4 𝐴𝐵
32frege64c 39062 . . 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 39059 . 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 1656   ∈ wcel 2166  [wsbc 3663 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-12 2222  ax-13 2391  ax-ext 2804  ax-frege1 38925  ax-frege2 38926  ax-frege8 38944  ax-frege58b 39036 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2813  df-cleq 2819  df-clel 2822  df-v 3417  df-sbc 3664 This theorem is referenced by:  frege66c  39064
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