Theorem eelT00

Description: An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	eelT00.1	$$⊢\top \to \phi$$
	eelT00.2	$$⊢\psi$$
	eelT00.3	$$⊢\chi$$
	eelT00.4	$$⊢\left(\phi \wedge \psi \wedge \chi \right)\to \theta$$
Assertion	eelT00	$$⊢\theta$$

Proof

Step	Hyp	Ref	Expression
1		eelT00.1	$$⊢\top \to \phi$$
2		eelT00.2	$$⊢\psi$$
3		eelT00.3	$$⊢\chi$$
4		eelT00.4	$$⊢\left(\phi \wedge \psi \wedge \chi \right)\to \theta$$
5		3anass	$$⊢\left(\top \wedge \psi \wedge \chi \right)\leftrightarrow \left(\top \wedge \left(\psi \wedge \chi \right)\right)$$
6		truan	$$⊢\left(\top \wedge \left(\psi \wedge \chi \right)\right)\leftrightarrow \left(\psi \wedge \chi \right)$$
7	5 6	bitri	$$⊢\left(\top \wedge \psi \wedge \chi \right)\leftrightarrow \left(\psi \wedge \chi \right)$$
8	1 4	syl3an1	$$⊢\left(\top \wedge \psi \wedge \chi \right)\to \theta$$
9	7 8	sylbir	$$⊢\left(\psi \wedge \chi \right)\to \theta$$
10	2 9	mpan	$$⊢\chi \to \theta$$
11	3 10	ax-mp	$$⊢\theta$$