Theorem frege66c

Description: Swap antecedents of frege65c . Proposition 66 of Frege1879 p. 54. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	frege59c.a	⊢ 𝐴 ∈ 𝐵
Assertion	frege66c	⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( ∀ 𝑥 ( 𝜒 → 𝜑 ) → ( [ 𝐴 / 𝑥 ] 𝜒 → [ 𝐴 / 𝑥 ] 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege59c.a	⊢ 𝐴 ∈ 𝐵
2	1	frege65c	⊢ ( ∀ 𝑥 ( 𝜒 → 𝜑 ) → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( [ 𝐴 / 𝑥 ] 𝜒 → [ 𝐴 / 𝑥 ] 𝜓 ) ) )
3		ax-frege8	⊢ ( ( ∀ 𝑥 ( 𝜒 → 𝜑 ) → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( [ 𝐴 / 𝑥 ] 𝜒 → [ 𝐴 / 𝑥 ] 𝜓 ) ) ) → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( ∀ 𝑥 ( 𝜒 → 𝜑 ) → ( [ 𝐴 / 𝑥 ] 𝜒 → [ 𝐴 / 𝑥 ] 𝜓 ) ) ) )
4	2 3	ax-mp	⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( ∀ 𝑥 ( 𝜒 → 𝜑 ) → ( [ 𝐴 / 𝑥 ] 𝜒 → [ 𝐴 / 𝑥 ] 𝜓 ) ) )