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Theorem frege65b 39045
 Description: A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2747 when the minor premise has a general context. Proposition 65 of [Frege1879] p. 53. In Frege care is taken to point out that the variables in the first clauses are independent of each other and of the final term so another valid translation could be : ⊢ (∀𝑥([𝑥 / 𝑎]𝜑 → [𝑥 / 𝑏]𝜓) → (∀𝑦([𝑦 / 𝑏]𝜓 → [𝑦 / 𝑐]𝜒) → ([𝑧 / 𝑎]𝜑 → [𝑧 / 𝑐]𝜒))). But that is perhaps too pedantic a translation for this exploration. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege65b (∀𝑥(𝜑𝜓) → (∀𝑥(𝜓𝜒) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜒)))

Proof of Theorem frege65b
StepHypRef Expression
1 sbim 2527 . . 3 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
2 frege64b 39044 . . 3 (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → (∀𝑥(𝜓𝜒) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜒)))
31, 2sylbi 209 . 2 ([𝑦 / 𝑥](𝜑𝜓) → (∀𝑥(𝜓𝜒) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜒)))
4 frege61b 39041 . 2 (([𝑦 / 𝑥](𝜑𝜓) → (∀𝑥(𝜓𝜒) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜒))) → (∀𝑥(𝜑𝜓) → (∀𝑥(𝜓𝜒) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜒))))
53, 4ax-mp 5 1 (∀𝑥(𝜑𝜓) → (∀𝑥(𝜓𝜒) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]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 38925  ax-frege2 38926  ax-frege8 38944  ax-frege58b 39036 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:  frege66b  39046
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