Theorem frege62c 42187

Description: A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2662 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 42183	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → [𝐴 / 𝑥](𝜑 → 𝜓))
3		sbcim1 3795	. . 3 ⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓))
4	2, 3	syl 17	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓))
5		ax-frege8 42071	. 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))

Syntax hints:   → wi 4  ∀wal 1539   ∈ wcel 2106  [wsbc 3739

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-frege8 42071  ax-frege58b 42163

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3447  df-sbc 3740

This theorem is referenced by:  frege63c  42188  frege64c  42189