Theorem frege62b 39041

Description: A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2746 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	⊢ ([𝑥 / 𝑦]𝜑 → (∀𝑦(𝜑 → 𝜓) → [𝑥 / 𝑦]𝜓))

Proof of Theorem frege62b

Step	Hyp	Ref	Expression
1		frege58bcor 39037	. 2 ⊢ (∀𝑦(𝜑 → 𝜓) → ([𝑥 / 𝑦]𝜑 → [𝑥 / 𝑦]𝜓))
2		ax-frege8 38943	. 2 ⊢ ((∀𝑦(𝜑 → 𝜓) → ([𝑥 / 𝑦]𝜑 → [𝑥 / 𝑦]𝜓)) → ([𝑥 / 𝑦]𝜑 → (∀𝑦(𝜑 → 𝜓) → [𝑥 / 𝑦]𝜓)))
3	1, 2	ax-mp 5	1 ⊢ ([𝑥 / 𝑦]𝜑 → (∀𝑦(𝜑 → 𝜓) → [𝑥 / 𝑦]𝜓))

Syntax hints:   → wi 4  ∀wal 1656  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-10 2194  ax-12 2222  ax-13 2391  ax-frege8 38943  ax-frege58b 39035

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-ex 1881  df-nf 1885  df-sb 2070

This theorem is referenced by:  frege63b  39042  frege64b  39043