Theorem frege61a 43203

Description: Lemma for frege65a 43207. Proposition 61 of [Frege1879] p. 52. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem frege61a

Step	Hyp	Ref	Expression
1		ax-frege58a 43199	. 2 ⊢ ((𝜓 ∧ 𝜒) → if-(𝜑, 𝜓, 𝜒))
2		frege9 43136	. 2 ⊢ (((𝜓 ∧ 𝜒) → if-(𝜑, 𝜓, 𝜒)) → ((if-(𝜑, 𝜓, 𝜒) → 𝜃) → ((𝜓 ∧ 𝜒) → 𝜃)))
3	1, 2	ax-mp 5	1 ⊢ ((if-(𝜑, 𝜓, 𝜒) → 𝜃) → ((𝜓 ∧ 𝜒) → 𝜃))

Syntax hints:   → wi 4   ∧ wa 395  if-wif 1059

This theorem was proved from axioms:  ax-mp 5  ax-frege1 43114  ax-frege2 43115  ax-frege8 43133  ax-frege58a 43199

This theorem is referenced by:  frege65a  43207