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Theorem frege65b 37224
 Description: A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2551 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 2383 . . 3 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
2 frege64b 37223 . . 3 (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → (∀𝑥(𝜓𝜒) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜒)))
31, 2sylbi 206 . 2 ([𝑦 / 𝑥](𝜑𝜓) → (∀𝑥(𝜓𝜒) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜒)))
4 frege61b 37220 . 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 1473  [wsb 1867 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234  ax-frege1 37104  ax-frege2 37105  ax-frege8 37123  ax-frege58b 37215 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868 This theorem is referenced by:  frege66b  37225
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