Theorem frege61a 45022

Description: Lemma for frege65a 45026. 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 45018	. 2 ⊢ ((𝜓 ∧ 𝜒) → if-(𝜑, 𝜓, 𝜒))
2		frege9 44955	. 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))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  if-wif 1078

This proof depends on axioms:  ax-mp 5  ax-frege1 44933  ax-frege2 44934  ax-frege8 44952  ax-frege58a 45018

This theorem is used by:  frege65a  45026