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Metamath Proof Explorer

Theorem ex-natded5.8-2

Description: A more efficient proof of Theorem 5.8 of Clemente p. 20. For a longer line-by-line translation, see ex-natded5.8 . (Contributed by Mario Carneiro, 9-Feb-2017) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ex-natded5.8.1	$⊢ φ \to ((ψ \land χ) \to \neg θ)$
	ex-natded5.8.2	$⊢ φ \to (τ \to θ)$
	ex-natded5.8.3	$⊢ φ \to χ$
	ex-natded5.8.4	$⊢ φ \to τ$
Assertion	ex-natded5.8-2	$⊢ φ \to \neg ψ$

Proof

Step	Hyp	Ref	Expression
1		ex-natded5.8.1	$⊢ φ \to ((ψ \land χ) \to \neg θ)$
2		ex-natded5.8.2	$⊢ φ \to (τ \to θ)$
3		ex-natded5.8.3	$⊢ φ \to χ$
4		ex-natded5.8.4	$⊢ φ \to τ$
5	4 2	mpd	$⊢ φ \to θ$
6	3 1	mpan2d	$⊢ φ \to (ψ \to \neg θ)$
7	5 6	mt2d	$⊢ φ \to \neg ψ$