Theorem frege67a 41493

Description: Lemma for frege68a 41494. Proposition 67 of [Frege1879] p. 54. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem frege67a

Step	Hyp	Ref	Expression
1		ax-frege58a 41483	. 2 ⊢ ((𝜓 ∧ 𝜒) → if-(𝜑, 𝜓, 𝜒))
2		frege7 41416	. 2 ⊢ (((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 5	1 ⊢ ((((𝜓 ∧ 𝜒) ↔ 𝜃) → (𝜃 → (𝜓 ∧ 𝜒))) → (((𝜓 ∧ 𝜒) ↔ 𝜃) → (𝜃 → if-(𝜑, 𝜓, 𝜒))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  if-wif 1060

This theorem was proved from axioms:  ax-mp 5  ax-frege1 41398  ax-frege2 41399  ax-frege58a 41483

This theorem is referenced by:  frege68a  41494