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	$$⊢\phi \to \left(\left(\psi \wedge \chi \right)\to \neg \theta \right)$$
	ex-natded5.8.2	$$⊢\phi \to \left(\tau \to \theta \right)$$
	ex-natded5.8.3	$$⊢\phi \to \chi$$
	ex-natded5.8.4	$$⊢\phi \to \tau$$
Assertion	ex-natded5.8-2	$$⊢\phi \to \neg \psi$$

Proof

Step	Hyp	Ref	Expression
1		ex-natded5.8.1	$$⊢\phi \to \left(\left(\psi \wedge \chi \right)\to \neg \theta \right)$$
2		ex-natded5.8.2	$$⊢\phi \to \left(\tau \to \theta \right)$$
3		ex-natded5.8.3	$$⊢\phi \to \chi$$
4		ex-natded5.8.4	$$⊢\phi \to \tau$$
5	4 2	mpd	$$⊢\phi \to \theta$$
6	3 1	mpan2d	$$⊢\phi \to \left(\psi \to \neg \theta \right)$$
7	5 6	mt2d	$$⊢\phi \to \neg \psi$$