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Theorem frege62c 43915
Description: A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2661 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 43911 . . 3 (∀𝑥(𝜑𝜓) → [𝐴 / 𝑥](𝜑𝜓))
3 sbcim1 3848 . . 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 43799 . 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 1535  wcel 2106  [wsbc 3791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-frege8 43799  ax-frege58b 43891
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-sbc 3792
This theorem is referenced by:  frege63c  43916  frege64c  43917
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