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Theorem frege64b 38520
Description: Lemma for frege65b 38521. 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 38518 . 2 ([𝑧 / 𝑦]𝜓 → (∀𝑦(𝜓𝜒) → [𝑧 / 𝑦]𝜒))
2 frege18 38429 . 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 1521  [wsb 1937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087  ax-13 2282  ax-frege1 38401  ax-frege2 38402  ax-frege8 38420  ax-frege58b 38512
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750  df-sb 1938
This theorem is referenced by:  frege65b  38521
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