Theorem frege60a

Description: Swap antecedents of ax-frege58a . Proposition 60 of Frege1879 p. 52. (Contributed by RP, 17-Apr-2020) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege60a	⊢ ( ( ( 𝜓 → ( 𝜒 → 𝜃 ) ) ∧ ( 𝜏 → ( 𝜂 → 𝜁 ) ) ) → ( if- ( 𝜑 , 𝜒 , 𝜂 ) → ( if- ( 𝜑 , 𝜓 , 𝜏 ) → if- ( 𝜑 , 𝜃 , 𝜁 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege58acor	⊢ ( ( ( 𝜓 → ( 𝜒 → 𝜃 ) ) ∧ ( 𝜏 → ( 𝜂 → 𝜁 ) ) ) → ( if- ( 𝜑 , 𝜓 , 𝜏 ) → if- ( 𝜑 , ( 𝜒 → 𝜃 ) , ( 𝜂 → 𝜁 ) ) ) )
2		ifpimim	⊢ ( if- ( 𝜑 , ( 𝜒 → 𝜃 ) , ( 𝜂 → 𝜁 ) ) → ( if- ( 𝜑 , 𝜒 , 𝜂 ) → if- ( 𝜑 , 𝜃 , 𝜁 ) ) )
3	1 2	syl6	⊢ ( ( ( 𝜓 → ( 𝜒 → 𝜃 ) ) ∧ ( 𝜏 → ( 𝜂 → 𝜁 ) ) ) → ( if- ( 𝜑 , 𝜓 , 𝜏 ) → ( if- ( 𝜑 , 𝜒 , 𝜂 ) → if- ( 𝜑 , 𝜃 , 𝜁 ) ) ) )
4		frege12	⊢ ( ( ( ( 𝜓 → ( 𝜒 → 𝜃 ) ) ∧ ( 𝜏 → ( 𝜂 → 𝜁 ) ) ) → ( if- ( 𝜑 , 𝜓 , 𝜏 ) → ( if- ( 𝜑 , 𝜒 , 𝜂 ) → if- ( 𝜑 , 𝜃 , 𝜁 ) ) ) ) → ( ( ( 𝜓 → ( 𝜒 → 𝜃 ) ) ∧ ( 𝜏 → ( 𝜂 → 𝜁 ) ) ) → ( if- ( 𝜑 , 𝜒 , 𝜂 ) → ( if- ( 𝜑 , 𝜓 , 𝜏 ) → if- ( 𝜑 , 𝜃 , 𝜁 ) ) ) ) )
5	3 4	ax-mp	⊢ ( ( ( 𝜓 → ( 𝜒 → 𝜃 ) ) ∧ ( 𝜏 → ( 𝜂 → 𝜁 ) ) ) → ( if- ( 𝜑 , 𝜒 , 𝜂 ) → ( if- ( 𝜑 , 𝜓 , 𝜏 ) → if- ( 𝜑 , 𝜃 , 𝜁 ) ) ) )