Theorem frege61b

Description: Lemma for frege65b . Proposition 61 of Frege1879 p. 52. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege61b	⊢ ( ( [ 𝑥 / 𝑦 ] 𝜑 → 𝜓 ) → ( ∀ 𝑦 𝜑 → 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		ax-frege58b	⊢ ( ∀ 𝑦 𝜑 → [ 𝑥 / 𝑦 ] 𝜑 )
2		frege9	⊢ ( ( ∀ 𝑦 𝜑 → [ 𝑥 / 𝑦 ] 𝜑 ) → ( ( [ 𝑥 / 𝑦 ] 𝜑 → 𝜓 ) → ( ∀ 𝑦 𝜑 → 𝜓 ) ) )
3	1 2	ax-mp	⊢ ( ( [ 𝑥 / 𝑦 ] 𝜑 → 𝜓 ) → ( ∀ 𝑦 𝜑 → 𝜓 ) )