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Theorem frege64c 37035
Description: Lemma for frege65c 37036. Proposition 64 of [Frege1879] p. 53. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
frege59c.a 𝐴𝐵
Assertion
Ref Expression
frege64c (([𝐶 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) → (∀𝑥(𝜓𝜒) → ([𝐶 / 𝑥]𝜑[𝐴 / 𝑥]𝜒)))

Proof of Theorem frege64c
StepHypRef Expression
1 frege59c.a . . 3 𝐴𝐵
21frege62c 37033 . 2 ([𝐴 / 𝑥]𝜓 → (∀𝑥(𝜓𝜒) → [𝐴 / 𝑥]𝜒))
3 frege18 36926 . 2 (([𝐴 / 𝑥]𝜓 → (∀𝑥(𝜓𝜒) → [𝐴 / 𝑥]𝜒)) → (([𝐶 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) → (∀𝑥(𝜓𝜒) → ([𝐶 / 𝑥]𝜑[𝐴 / 𝑥]𝜒))))
42, 3ax-mp 5 1 (([𝐶 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) → (∀𝑥(𝜓𝜒) → ([𝐶 / 𝑥]𝜑[𝐴 / 𝑥]𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1473  wcel 1977  [wsbc 3402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234  ax-ext 2590  ax-frege1 36898  ax-frege2 36899  ax-frege8 36917  ax-frege58b 37009
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175  df-sbc 3403
This theorem is referenced by:  frege65c  37036
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