Theorem frege68a 41383

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 41369	. 2 ⊢ (((𝜓 ∧ 𝜒) ↔ 𝜃) → (𝜃 → (𝜓 ∧ 𝜒)))
2		frege67a 41382	. 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 205   ∧ wa 395  if-wif 1059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege1 41287  ax-frege2 41288  ax-frege8 41306  ax-frege52a 41354  ax-frege58a 41372

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ifp 1060  df-tru 1542  df-fal 1552

This theorem is referenced by: (None)