Theorem frege64c

Description: Lemma for frege65c . Proposition 64 of Frege1879 p. 53. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	frege59c.a	⊢ 𝐴 ∈ 𝐵
Assertion	frege64c	⊢ ( ( [ 𝐶 / 𝑥 ] 𝜑 → [ 𝐴 / 𝑥 ] 𝜓 ) → ( ∀ 𝑥 ( 𝜓 → 𝜒 ) → ( [ 𝐶 / 𝑥 ] 𝜑 → [ 𝐴 / 𝑥 ] 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege59c.a	⊢ 𝐴 ∈ 𝐵
2	1	frege62c	⊢ ( [ 𝐴 / 𝑥 ] 𝜓 → ( ∀ 𝑥 ( 𝜓 → 𝜒 ) → [ 𝐴 / 𝑥 ] 𝜒 ) )
3		frege18	⊢ ( ( [ 𝐴 / 𝑥 ] 𝜓 → ( ∀ 𝑥 ( 𝜓 → 𝜒 ) → [ 𝐴 / 𝑥 ] 𝜒 ) ) → ( ( [ 𝐶 / 𝑥 ] 𝜑 → [ 𝐴 / 𝑥 ] 𝜓 ) → ( ∀ 𝑥 ( 𝜓 → 𝜒 ) → ( [ 𝐶 / 𝑥 ] 𝜑 → [ 𝐴 / 𝑥 ] 𝜒 ) ) ) )
4	2 3	ax-mp	⊢ ( ( [ 𝐶 / 𝑥 ] 𝜑 → [ 𝐴 / 𝑥 ] 𝜓 ) → ( ∀ 𝑥 ( 𝜓 → 𝜒 ) → ( [ 𝐶 / 𝑥 ] 𝜑 → [ 𝐴 / 𝑥 ] 𝜒 ) ) )