Theorem frege63b

Description: Lemma for frege91 . Proposition 63 of Frege1879 p. 52. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege63b	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 → ( 𝜓 → ( ∀ 𝑥 ( 𝜑 → 𝜒 ) → [ 𝑦 / 𝑥 ] 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege62b	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 → ( ∀ 𝑥 ( 𝜑 → 𝜒 ) → [ 𝑦 / 𝑥 ] 𝜒 ) )
2		frege24	⊢ ( ( [ 𝑦 / 𝑥 ] 𝜑 → ( ∀ 𝑥 ( 𝜑 → 𝜒 ) → [ 𝑦 / 𝑥 ] 𝜒 ) ) → ( [ 𝑦 / 𝑥 ] 𝜑 → ( 𝜓 → ( ∀ 𝑥 ( 𝜑 → 𝜒 ) → [ 𝑦 / 𝑥 ] 𝜒 ) ) ) )
3	1 2	ax-mp	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 → ( 𝜓 → ( ∀ 𝑥 ( 𝜑 → 𝜒 ) → [ 𝑦 / 𝑥 ] 𝜒 ) ) )