Theorem frege66a

Description: Swap antecedents of frege65a . Proposition 66 of Frege1879 p. 54. (Contributed by RP, 17-Apr-2020) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege66a	⊢ ( ( ( 𝜒 → 𝜃 ) ∧ ( 𝜂 → 𝜁 ) ) → ( ( ( 𝜓 → 𝜒 ) ∧ ( 𝜏 → 𝜂 ) ) → ( if- ( 𝜑 , 𝜓 , 𝜏 ) → if- ( 𝜑 , 𝜃 , 𝜁 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege65a	⊢ ( ( ( 𝜓 → 𝜒 ) ∧ ( 𝜏 → 𝜂 ) ) → ( ( ( 𝜒 → 𝜃 ) ∧ ( 𝜂 → 𝜁 ) ) → ( if- ( 𝜑 , 𝜓 , 𝜏 ) → if- ( 𝜑 , 𝜃 , 𝜁 ) ) ) )
2		ax-frege8	⊢ ( ( ( ( 𝜓 → 𝜒 ) ∧ ( 𝜏 → 𝜂 ) ) → ( ( ( 𝜒 → 𝜃 ) ∧ ( 𝜂 → 𝜁 ) ) → ( if- ( 𝜑 , 𝜓 , 𝜏 ) → if- ( 𝜑 , 𝜃 , 𝜁 ) ) ) ) → ( ( ( 𝜒 → 𝜃 ) ∧ ( 𝜂 → 𝜁 ) ) → ( ( ( 𝜓 → 𝜒 ) ∧ ( 𝜏 → 𝜂 ) ) → ( if- ( 𝜑 , 𝜓 , 𝜏 ) → if- ( 𝜑 , 𝜃 , 𝜁 ) ) ) ) )
3	1 2	ax-mp	⊢ ( ( ( 𝜒 → 𝜃 ) ∧ ( 𝜂 → 𝜁 ) ) → ( ( ( 𝜓 → 𝜒 ) ∧ ( 𝜏 → 𝜂 ) ) → ( if- ( 𝜑 , 𝜓 , 𝜏 ) → if- ( 𝜑 , 𝜃 , 𝜁 ) ) ) )