Theorem frege67a

Description: Lemma for frege68a . Proposition 67 of Frege1879 p. 54. (Contributed by RP, 17-Apr-2020) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege67a	⊢ ( ( ( ( 𝜓 ∧ 𝜒 ) ↔ 𝜃 ) → ( 𝜃 → ( 𝜓 ∧ 𝜒 ) ) ) → ( ( ( 𝜓 ∧ 𝜒 ) ↔ 𝜃 ) → ( 𝜃 → if- ( 𝜑 , 𝜓 , 𝜒 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		ax-frege58a	⊢ ( ( 𝜓 ∧ 𝜒 ) → if- ( 𝜑 , 𝜓 , 𝜒 ) )
2		frege7	⊢ ( ( ( 𝜓 ∧ 𝜒 ) → if- ( 𝜑 , 𝜓 , 𝜒 ) ) → ( ( ( ( 𝜓 ∧ 𝜒 ) ↔ 𝜃 ) → ( 𝜃 → ( 𝜓 ∧ 𝜒 ) ) ) → ( ( ( 𝜓 ∧ 𝜒 ) ↔ 𝜃 ) → ( 𝜃 → if- ( 𝜑 , 𝜓 , 𝜒 ) ) ) ) )
3	1 2	ax-mp	⊢ ( ( ( ( 𝜓 ∧ 𝜒 ) ↔ 𝜃 ) → ( 𝜃 → ( 𝜓 ∧ 𝜒 ) ) ) → ( ( ( 𝜓 ∧ 𝜒 ) ↔ 𝜃 ) → ( 𝜃 → if- ( 𝜑 , 𝜓 , 𝜒 ) ) ) )