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Metamath Proof Explorer

Theorem ex-natded5.13-2

Description: A more efficient proof of Theorem 5.13 of Clemente p. 20. Compare with ex-natded5.13 . (Contributed by Mario Carneiro, 9-Feb-2017) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ex-natded5.13.1	$⊢ φ \to (ψ \lor χ)$
	ex-natded5.13.2	$⊢ φ \to (ψ \to θ)$
	ex-natded5.13.3	$⊢ φ \to (\neg τ \to \neg χ)$
Assertion	ex-natded5.13-2	$⊢ φ \to (θ \lor τ)$

Proof

Step	Hyp	Ref	Expression
1		ex-natded5.13.1	$⊢ φ \to (ψ \lor χ)$
2		ex-natded5.13.2	$⊢ φ \to (ψ \to θ)$
3		ex-natded5.13.3	$⊢ φ \to (\neg τ \to \neg χ)$
4	3	con4d	$⊢ φ \to (χ \to τ)$
5	2 4	orim12d	$⊢ φ \to ((ψ \lor χ) \to (θ \lor τ))$
6	1 5	mpd	$⊢ φ \to (θ \lor τ)$