Theorem frege62b 37221

Description: A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2551 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 37217	. 2 ⊢ (∀𝑦(𝜑 → 𝜓) → ([𝑥 / 𝑦]𝜑 → [𝑥 / 𝑦]𝜓))
2		ax-frege8 37123	. 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 1473  [wsb 1867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234  ax-frege8 37123  ax-frege58b 37215

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868

This theorem is referenced by:  frege63b  37222  frege64b  37223