Theorem orim12dALT

Description: Alternate proof of orim12d which does not depend on df-an . This is an illustration of the conservativity of definitions (definitions do not permit to prove additional theorems whose statements do not contain the defined symbol). (Contributed by Wolf Lammen, 8-Aug-2022) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	orim12dALT.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
	orim12dALT.2	$$⊢\phi \to \left(\theta \to \tau \right)$$
Assertion	orim12dALT	$$⊢\phi \to \left(\left(\psi \vee \theta \right)\to \left(\chi \vee \tau \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		orim12dALT.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
2		orim12dALT.2	$$⊢\phi \to \left(\theta \to \tau \right)$$
3		pm2.53	$$⊢\left(\psi \vee \theta \right)\to \left(\neg \psi \to \theta \right)$$
4	1	con3d	$$⊢\phi \to \left(\neg \chi \to \neg \psi \right)$$
5	4 2	imim12d	$$⊢\phi \to \left(\left(\neg \psi \to \theta \right)\to \left(\neg \chi \to \tau \right)\right)$$
6		pm2.54	$$⊢\left(\neg \chi \to \tau \right)\to \left(\chi \vee \tau \right)$$
7	3 5 6	syl56	$$⊢\phi \to \left(\left(\psi \vee \theta \right)\to \left(\chi \vee \tau \right)\right)$$