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Theorem frege63c 40798
 Description: Analogue of frege63b 40780. 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 40797 . 2 ([𝐴 / 𝑥]𝜑 → (∀𝑥(𝜑𝜒) → [𝐴 / 𝑥]𝜒))
3 frege24 40687 . 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 1536   ∈ wcel 2111  [wsbc 3722 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2770  ax-frege1 40662  ax-frege2 40663  ax-frege8 40681  ax-frege58b 40773 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3444  df-sbc 3723 This theorem is referenced by:  frege91  40826
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