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Theorem frege63c 43142
Description: Analogue of frege63b 43124. Proposition 63 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
frege59c.a 𝐴𝐵
Assertion
Ref Expression
frege63c ([𝐴 / 𝑥]𝜑 → (𝜓 → (∀𝑥(𝜑𝜒) → [𝐴 / 𝑥]𝜒)))

Proof of Theorem frege63c
StepHypRef Expression
1 frege59c.a . . 3 𝐴𝐵
21frege62c 43141 . 2 ([𝐴 / 𝑥]𝜑 → (∀𝑥(𝜑𝜒) → [𝐴 / 𝑥]𝜒))
3 frege24 43031 . 2 (([𝐴 / 𝑥]𝜑 → (∀𝑥(𝜑𝜒) → [𝐴 / 𝑥]𝜒)) → ([𝐴 / 𝑥]𝜑 → (𝜓 → (∀𝑥(𝜑𝜒) → [𝐴 / 𝑥]𝜒))))
42, 3ax-mp 5 1 ([𝐴 / 𝑥]𝜑 → (𝜓 → (∀𝑥(𝜑𝜒) → [𝐴 / 𝑥]𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1538  wcel 2105  [wsbc 3777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-frege1 43006  ax-frege2 43007  ax-frege8 43025  ax-frege58b 43117
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-sbc 3778
This theorem is referenced by:  frege91  43170
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