Theorem frege61a

Description: Lemma for frege65a . Proposition 61 of Frege1879 p. 52. (Contributed by RP, 17-Apr-2020) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege61a	⊢ ( ( if- ( 𝜑 , 𝜓 , 𝜒 ) → 𝜃 ) → ( ( 𝜓 ∧ 𝜒 ) → 𝜃 ) )

Proof

Step	Hyp	Ref	Expression
1		ax-frege58a	⊢ ( ( 𝜓 ∧ 𝜒 ) → if- ( 𝜑 , 𝜓 , 𝜒 ) )
2		frege9	⊢ ( ( ( 𝜓 ∧ 𝜒 ) → if- ( 𝜑 , 𝜓 , 𝜒 ) ) → ( ( if- ( 𝜑 , 𝜓 , 𝜒 ) → 𝜃 ) → ( ( 𝜓 ∧ 𝜒 ) → 𝜃 ) ) )
3	1 2	ax-mp	⊢ ( ( if- ( 𝜑 , 𝜓 , 𝜒 ) → 𝜃 ) → ( ( 𝜓 ∧ 𝜒 ) → 𝜃 ) )