Theorem frege65c 43941

Description: A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2663 when the minor premise has a general context. Proposition 65 of [Frege1879] p. 53. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
frege59c.a	⊢ 𝐴 ∈ 𝐵

Assertion

Ref	Expression
frege65c	⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥(𝜓 → 𝜒) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜒)))

Proof of Theorem frege65c

Step	Hyp	Ref	Expression
1		sbcim1 3842	. . 3 ⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓))
2		frege59c.a	. . . 4 ⊢ 𝐴 ∈ 𝐵
3	2	frege64c 43940	. . 3 ⊢ (([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓) → (∀𝑥(𝜓 → 𝜒) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜒)))
4	1, 3	syl 17	. 2 ⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) → (∀𝑥(𝜓 → 𝜒) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜒)))
5	2	frege61c 43937	. 2 ⊢ (([𝐴 / 𝑥](𝜑 → 𝜓) → (∀𝑥(𝜓 → 𝜒) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜒))) → (∀𝑥(𝜑 → 𝜓) → (∀𝑥(𝜓 → 𝜒) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜒))))
6	4, 5	ax-mp 5	1 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥(𝜓 → 𝜒) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜒)))

Syntax hints:   → wi 4  ∀wal 1538   ∈ wcel 2108  [wsbc 3788

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-frege1 43803  ax-frege2 43804  ax-frege8 43822  ax-frege58b 43914

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-sbc 3789

This theorem is referenced by:  frege66c  43942