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	$$⊢\left(\left(\psi \wedge \chi \right)\leftrightarrow \theta \right)\to \left(\theta \to if-\left(\phi ,\psi ,\chi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege57aid	$$⊢\left(\left(\psi \wedge \chi \right)\leftrightarrow \theta \right)\to \left(\theta \to \left(\psi \wedge \chi \right)\right)$$
2		frege67a	$$⊢\left(\left(\left(\psi \wedge \chi \right)\leftrightarrow \theta \right)\to \left(\theta \to \left(\psi \wedge \chi \right)\right)\right)\to \left(\left(\left(\psi \wedge \chi \right)\leftrightarrow \theta \right)\to \left(\theta \to if-\left(\phi ,\psi ,\chi \right)\right)\right)$$
3	1 2	ax-mp	$$⊢\left(\left(\psi \wedge \chi \right)\leftrightarrow \theta \right)\to \left(\theta \to if-\left(\phi ,\psi ,\chi \right)\right)$$