Metamath Proof Explorer


Theorem frege68a

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.)

Ref Expression
Assertion frege68a ( ( ( 𝜓𝜒 ) ↔ 𝜃 ) → ( 𝜃 → if- ( 𝜑 , 𝜓 , 𝜒 ) ) )

Proof

Step Hyp Ref Expression
1 frege57aid ( ( ( 𝜓𝜒 ) ↔ 𝜃 ) → ( 𝜃 → ( 𝜓𝜒 ) ) )
2 frege67a ( ( ( ( 𝜓𝜒 ) ↔ 𝜃 ) → ( 𝜃 → ( 𝜓𝜒 ) ) ) → ( ( ( 𝜓𝜒 ) ↔ 𝜃 ) → ( 𝜃 → if- ( 𝜑 , 𝜓 , 𝜒 ) ) ) )
3 1 2 ax-mp ( ( ( 𝜓𝜒 ) ↔ 𝜃 ) → ( 𝜃 → if- ( 𝜑 , 𝜓 , 𝜒 ) ) )