Theorem frege62b 42648

Description: A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2658 when the minor premise has a particular context. Proposition 62 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege62b	⊢ ([𝑦 / 𝑥]𝜑 → (∀𝑥(𝜑 → 𝜓) → [𝑦 / 𝑥]𝜓))

Proof of Theorem frege62b

Step	Hyp	Ref	Expression
1		frege58bcor 42644	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
2		ax-frege8 42550	. 2 ⊢ ((∀𝑥(𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) → ([𝑦 / 𝑥]𝜑 → (∀𝑥(𝜑 → 𝜓) → [𝑦 / 𝑥]𝜓)))
3	1, 2	ax-mp 5	1 ⊢ ([𝑦 / 𝑥]𝜑 → (∀𝑥(𝜑 → 𝜓) → [𝑦 / 𝑥]𝜓))

Syntax hints:   → wi 4  ∀wal 1539  [wsb 2067

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-10 2137  ax-12 2171  ax-frege8 42550  ax-frege58b 42642

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ex 1782  df-nf 1786  df-sb 2068

This theorem is referenced by:  frege63b  42649  frege64b  42650