Theorem eelTTT

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

	Ref	Expression
Hypotheses	eelTTT.1	|- ( T. -> ph )
	eelTTT.2	|- ( T. -> ps )
	eelTTT.3	|- ( T. -> ch )
	eelTTT.4	|- ( ( ph /\ ps /\ ch ) -> th )
Assertion	eelTTT	|- th

Proof

Step	Hyp	Ref	Expression
1		eelTTT.1	|- ( T. -> ph )
2		eelTTT.2	|- ( T. -> ps )
3		eelTTT.3	|- ( T. -> ch )
4		eelTTT.4	|- ( ( ph /\ ps /\ ch ) -> th )
5		truan	|- ( ( T. /\ ch ) <-> ch )
6		3anass	|- ( ( T. /\ ps /\ ch ) <-> ( T. /\ ( ps /\ ch ) ) )
7		truan	|- ( ( T. /\ ( ps /\ ch ) ) <-> ( ps /\ ch ) )
8	6 7	bitri	|- ( ( T. /\ ps /\ ch ) <-> ( ps /\ ch ) )
9	1 4	syl3an1	|- ( ( T. /\ ps /\ ch ) -> th )
10	8 9	sylbir	|- ( ( ps /\ ch ) -> th )
11	2 10	sylan	|- ( ( T. /\ ch ) -> th )
12	5 11	sylbir	|- ( ch -> th )
13	3 12	syl	|- ( T. -> th )
14	13	mptru	|- th