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Theorem frege65b 40542
Description: A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2749 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 2312 . . 3 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
2 frege64b 40541 . . 3 (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → (∀𝑥(𝜓𝜒) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜒)))
31, 2sylbi 220 . 2 ([𝑦 / 𝑥](𝜑𝜓) → (∀𝑥(𝜓𝜒) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜒)))
4 frege61b 40538 . 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 1536  [wsb 2069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-12 2178  ax-frege1 40422  ax-frege2 40423  ax-frege8 40441  ax-frege58b 40533
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070
This theorem is referenced by:  frege66b  40543
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