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Theorem frege64b 39043
 Description: Lemma for frege65b 39044. Proposition 64 of [Frege1879] p. 53. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege64b (([𝑥 / 𝑦]𝜑 → [𝑧 / 𝑦]𝜓) → (∀𝑦(𝜓𝜒) → ([𝑥 / 𝑦]𝜑 → [𝑧 / 𝑦]𝜒)))

Proof of Theorem frege64b
StepHypRef Expression
1 frege62b 39041 . 2 ([𝑧 / 𝑦]𝜓 → (∀𝑦(𝜓𝜒) → [𝑧 / 𝑦]𝜒))
2 frege18 38952 . 2 (([𝑧 / 𝑦]𝜓 → (∀𝑦(𝜓𝜒) → [𝑧 / 𝑦]𝜒)) → (([𝑥 / 𝑦]𝜑 → [𝑧 / 𝑦]𝜓) → (∀𝑦(𝜓𝜒) → ([𝑥 / 𝑦]𝜑 → [𝑧 / 𝑦]𝜒))))
31, 2ax-mp 5 1 (([𝑥 / 𝑦]𝜑 → [𝑧 / 𝑦]𝜓) → (∀𝑦(𝜓𝜒) → ([𝑥 / 𝑦]𝜑 → [𝑧 / 𝑦]𝜒)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1656  [wsb 2069 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-10 2194  ax-12 2222  ax-13 2391  ax-frege1 38924  ax-frege2 38925  ax-frege8 38943  ax-frege58b 39035 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-ex 1881  df-nf 1885  df-sb 2070 This theorem is referenced by:  frege65b  39044
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