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Metamath Proof Explorer

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	$⊢ φ \to (ψ \to χ)$
	orim12dALT.2	$⊢ φ \to (θ \to τ)$
Assertion	orim12dALT	$⊢ φ \to ((ψ \lor θ) \to (χ \lor τ))$

Proof

Step	Hyp	Ref	Expression
1		orim12dALT.1	$⊢ φ \to (ψ \to χ)$
2		orim12dALT.2	$⊢ φ \to (θ \to τ)$
3		pm2.53	$⊢ (ψ \lor θ) \to (\neg ψ \to θ)$
4	1	con3d	$⊢ φ \to (\neg χ \to \neg ψ)$
5	4 2	imim12d	$⊢ φ \to ((\neg ψ \to θ) \to (\neg χ \to τ))$
6		pm2.54	$⊢ (\neg χ \to τ) \to (χ \lor τ)$
7	3 5 6	syl56	$⊢ φ \to ((ψ \lor θ) \to (χ \lor τ))$