Theorem frege62b 44484

Description: A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2690 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 44480	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
2		ax-frege8 44386	. 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 1559  [wsb 2091

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-10 2176  ax-12 2213  ax-frege8 44386  ax-frege58b 44478

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1801  df-nf 1805  df-sb 2092

This theorem is referenced by:  frege63b  44485  frege64b  44486