Theorem orim12da

Description: Deduce a disjunction from another one. Variation on orim12d . (Contributed by Thierry Arnoux, 18-May-2025)

	Ref	Expression
Hypotheses	orim12da.1	$$⊢\left(\phi \wedge \psi \right)\to \theta$$
	orim12da.2	$$⊢\left(\phi \wedge \chi \right)\to \tau$$
	orim12da.3	$$⊢\phi \to \left(\psi \vee \chi \right)$$
Assertion	orim12da	$$⊢\phi \to \left(\theta \vee \tau \right)$$

Proof

Step	Hyp	Ref	Expression
1		orim12da.1	$$⊢\left(\phi \wedge \psi \right)\to \theta$$
2		orim12da.2	$$⊢\left(\phi \wedge \chi \right)\to \tau$$
3		orim12da.3	$$⊢\phi \to \left(\psi \vee \chi \right)$$
4	1	ex	$$⊢\phi \to \left(\psi \to \theta \right)$$
5	2	ex	$$⊢\phi \to \left(\chi \to \tau \right)$$
6	4 5	orim12d	$$⊢\phi \to \left(\left(\psi \vee \chi \right)\to \left(\theta \vee \tau \right)\right)$$
7	3 6	mpd	$$⊢\phi \to \left(\theta \vee \tau \right)$$