Theorem frege65b 37725

Description: A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2562 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(𝜑 → 𝜓) → (∀𝑥(𝜓 → 𝜒) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜒)))

Proof of Theorem frege65b

Step	Hyp	Ref	Expression
1		sbim 2394	. . 3 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
2		frege64b 37724	. . 3 ⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → (∀𝑥(𝜓 → 𝜒) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜒)))
3	1, 2	sylbi 207	. 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) → (∀𝑥(𝜓 → 𝜒) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜒)))
4		frege61b 37721	. 2 ⊢ (([𝑦 / 𝑥](𝜑 → 𝜓) → (∀𝑥(𝜓 → 𝜒) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜒))) → (∀𝑥(𝜑 → 𝜓) → (∀𝑥(𝜓 → 𝜒) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜒))))
5	3, 4	ax-mp 5	1 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥(𝜓 → 𝜒) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜒)))

Syntax hints:   → wi 4  ∀wal 1478  [wsb 1877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-12 2044  ax-13 2245  ax-frege1 37605  ax-frege2 37606  ax-frege8 37624  ax-frege58b 37716

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702  df-nf 1707  df-sb 1878

This theorem is referenced by:  frege66b  37726