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Theorem frege64b 40133
Description: Lemma for frege65b 40134. 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 40131 . 2 ([𝑧 / 𝑦]𝜓 → (∀𝑦(𝜓𝜒) → [𝑧 / 𝑦]𝜒))
2 frege18 40042 . 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 1526  [wsb 2060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-10 2136  ax-12 2167  ax-frege1 40014  ax-frege2 40015  ax-frege8 40033  ax-frege58b 40125
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ex 1772  df-nf 1776  df-sb 2061
This theorem is referenced by:  frege65b  40134
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