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Theorem frege63c 39059
Description: Analogue of frege63b 39041. 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 39058 . 2 ([𝐴 / 𝑥]𝜑 → (∀𝑥(𝜑𝜒) → [𝐴 / 𝑥]𝜒))
3 frege24 38948 . 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 1654  wcel 2164  [wsbc 3662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-12 2220  ax-13 2389  ax-ext 2803  ax-frege1 38923  ax-frege2 38924  ax-frege8 38942  ax-frege58b 39034
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-v 3416  df-sbc 3663
This theorem is referenced by:  frege91  39087
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