Theorem frege68a 39021

Description: Combination of applying a definition and applying it to a specific instance. Proposition 68 of [Frege1879] p. 54. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege68a	⊢ (((𝜓 ∧ 𝜒) ↔ 𝜃) → (𝜃 → if-(𝜑, 𝜓, 𝜒)))

Proof of Theorem frege68a

Step	Hyp	Ref	Expression
1		frege57aid 39007	. 2 ⊢ (((𝜓 ∧ 𝜒) ↔ 𝜃) → (𝜃 → (𝜓 ∧ 𝜒)))
2		frege67a 39020	. 2 ⊢ ((((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 → if-(𝜑, 𝜓, 𝜒)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386  if-wif 1091

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege1 38925  ax-frege2 38926  ax-frege8 38944  ax-frege52a 38992  ax-frege58a 39010

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-ifp 1092  df-tru 1662  df-fal 1672

This theorem is referenced by: (None)