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Theorem frege64b 37022
Description: Lemma for frege65b 37023. 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 37020 . 2 ([𝑧 / 𝑦]𝜓 → (∀𝑦(𝜓𝜒) → [𝑧 / 𝑦]𝜒))
2 frege18 36931 . 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 1472  [wsb 1865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-12 2031  ax-13 2228  ax-frege1 36903  ax-frege2 36904  ax-frege8 36922  ax-frege58b 37014
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866
This theorem is referenced by:  frege65b  37023
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