Theorem eelTTT 38931

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

Hypotheses

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

Assertion

Ref	Expression
eelTTT	|- th

Proof of Theorem eelTTT

Step	Hyp	Ref	Expression
1		eelTTT.3	. . 3 |- ( T. -> ch )
2		truan 1501	. . . 4 |- ( ( T. /\ ch ) <-> ch )
3		eelTTT.2	. . . . 5 |- ( T. -> ps )
4		3anass 1042	. . . . . . 7 |- ( ( T. /\ ps /\ ch ) <-> ( T. /\ ( ps /\ ch ) ) )
5		truan 1501	. . . . . . 7 |- ( ( T. /\ ( ps /\ ch ) ) <-> ( ps /\ ch ) )
6	4, 5	bitri 264	. . . . . 6 |- ( ( T. /\ ps /\ ch ) <-> ( ps /\ ch ) )
7		eelTTT.1	. . . . . . 7 |- ( T. -> ph )
8		eelTTT.4	. . . . . . 7 |- ( ( ph /\ ps /\ ch ) -> th )
9	7, 8	syl3an1 1359	. . . . . 6 |- ( ( T. /\ ps /\ ch ) -> th )
10	6, 9	sylbir 225	. . . . 5 |- ( ( ps /\ ch ) -> th )
11	3, 10	sylan 488	. . . 4 |- ( ( T. /\ ch ) -> th )
12	2, 11	sylbir 225	. . 3 |- ( ch -> th )
13	1, 12	syl 17	. 2 |- ( T. -> th )
14	13	trud 1493	1 |- th

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   T. wtru 1484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-an 386  df-3an 1039  df-tru 1486

This theorem is referenced by: (None)