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Theorem frege62c 43592
Description: A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2652 when the minor premise has a particular context. Proposition 62 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
frege59c.a 𝐴𝐵
Assertion
Ref Expression
frege62c ([𝐴 / 𝑥]𝜑 → (∀𝑥(𝜑𝜓) → [𝐴 / 𝑥]𝜓))

Proof of Theorem frege62c
StepHypRef Expression
1 frege59c.a . . . 4 𝐴𝐵
21frege58c 43588 . . 3 (∀𝑥(𝜑𝜓) → [𝐴 / 𝑥](𝜑𝜓))
3 sbcim1 3833 . . 3 ([𝐴 / 𝑥](𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))
42, 3syl 17 . 2 (∀𝑥(𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))
5 ax-frege8 43476 . 2 ((∀𝑥(𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)) → ([𝐴 / 𝑥]𝜑 → (∀𝑥(𝜑𝜓) → [𝐴 / 𝑥]𝜓)))
64, 5ax-mp 5 1 ([𝐴 / 𝑥]𝜑 → (∀𝑥(𝜑𝜓) → [𝐴 / 𝑥]𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1532  wcel 2099  [wsbc 3776
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-8 2101  ax-9 2109  ax-ext 2697  ax-frege8 43476  ax-frege58b 43568
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-v 3464  df-sbc 3777
This theorem is referenced by:  frege63c  43593  frege64c  43594
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