Theorem frege64a

Description: Lemma for frege65a . Proposition 64 of Frege1879 p. 53. (Contributed by RP, 17-Apr-2020) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege64a	⊢ ( ( if- ( 𝜑 , 𝜓 , 𝜏 ) → if- ( 𝜎 , 𝜒 , 𝜂 ) ) → ( ( ( 𝜒 → 𝜃 ) ∧ ( 𝜂 → 𝜁 ) ) → ( if- ( 𝜑 , 𝜓 , 𝜏 ) → if- ( 𝜎 , 𝜃 , 𝜁 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege62a	⊢ ( if- ( 𝜎 , 𝜒 , 𝜂 ) → ( ( ( 𝜒 → 𝜃 ) ∧ ( 𝜂 → 𝜁 ) ) → if- ( 𝜎 , 𝜃 , 𝜁 ) ) )
2		frege18	⊢ ( ( if- ( 𝜎 , 𝜒 , 𝜂 ) → ( ( ( 𝜒 → 𝜃 ) ∧ ( 𝜂 → 𝜁 ) ) → if- ( 𝜎 , 𝜃 , 𝜁 ) ) ) → ( ( if- ( 𝜑 , 𝜓 , 𝜏 ) → if- ( 𝜎 , 𝜒 , 𝜂 ) ) → ( ( ( 𝜒 → 𝜃 ) ∧ ( 𝜂 → 𝜁 ) ) → ( if- ( 𝜑 , 𝜓 , 𝜏 ) → if- ( 𝜎 , 𝜃 , 𝜁 ) ) ) ) )
3	1 2	ax-mp	⊢ ( ( if- ( 𝜑 , 𝜓 , 𝜏 ) → if- ( 𝜎 , 𝜒 , 𝜂 ) ) → ( ( ( 𝜒 → 𝜃 ) ∧ ( 𝜂 → 𝜁 ) ) → ( if- ( 𝜑 , 𝜓 , 𝜏 ) → if- ( 𝜎 , 𝜃 , 𝜁 ) ) ) )