Theorem frege62c 44711

Description: A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2692 when the minor premise has a particular context. Proposition 62 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
frege59c.a	⊢ 𝐴 ∈ 𝐵

Assertion

Ref	Expression
frege62c	⊢ ([𝐴 / 𝑥]𝜑 → (∀𝑥(𝜑 → 𝜓) → [𝐴 / 𝑥]𝜓))

Proof of Theorem frege62c

Step	Hyp	Ref	Expression
1		frege59c.a	. . . 4 ⊢ 𝐴 ∈ 𝐵
2	1	frege58c 44707	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → [𝐴 / 𝑥](𝜑 → 𝜓))
3		sbcim1 3799	. . 3 ⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓))
4	2, 3	syl 18	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓))
5		ax-frege8 44595	. 2 ⊢ ((∀𝑥(𝜑 → 𝜓) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓)) → ([𝐴 / 𝑥]𝜑 → (∀𝑥(𝜑 → 𝜓) → [𝐴 / 𝑥]𝜓)))
6	4, 5	ax-mp 5	1 ⊢ ([𝐴 / 𝑥]𝜑 → (∀𝑥(𝜑 → 𝜓) → [𝐴 / 𝑥]𝜓))

This proof depends on syntax axioms:   → wi 4  ∀wal 1568   ∈ wcel 2146  [wsbc 3746

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-frege8 44595  ax-frege58b 44687

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-sbc 3747

This theorem is used by:  frege63c  44712  frege64c  44713