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Theorem frege62c 42187
Description: A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2662 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 42183 . . 3 (∀𝑥(𝜑𝜓) → [𝐴 / 𝑥](𝜑𝜓))
3 sbcim1 3795 . . 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 42071 . 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 1539  wcel 2106  [wsbc 3739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-frege8 42071  ax-frege58b 42163
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3447  df-sbc 3740
This theorem is referenced by:  frege63c  42188  frege64c  42189
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