Theorem frege10

Description: Result commuting antecedents within an antecedent. Proposition 10 of Frege1879 p. 36. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege10	⊢ ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → 𝜃 ) → ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) → 𝜃 ) )

Proof

Step	Hyp	Ref	Expression
1		ax-frege8	⊢ ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) → ( 𝜑 → ( 𝜓 → 𝜒 ) ) )
2		frege9	⊢ ( ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) → ( 𝜑 → ( 𝜓 → 𝜒 ) ) ) → ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → 𝜃 ) → ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) → 𝜃 ) ) )
3	1 2	ax-mp	⊢ ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → 𝜃 ) → ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) → 𝜃 ) )