Theorem frege62b

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

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

Proof

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