Theorem eelT00 38930

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

Hypotheses

Ref	Expression
eelT00.1	|- ( T. -> ph )
eelT00.2	|- ps
eelT00.3	|- ch
eelT00.4	|- ( ( ph /\ ps /\ ch ) -> th )

Assertion

Ref	Expression
eelT00	|- th

Proof of Theorem eelT00

Step	Hyp	Ref	Expression
1		eelT00.3	. 2 |- ch
2		eelT00.2	. . 3 |- ps
3		3anass 1042	. . . . 5 |- ( ( T. /\ ps /\ ch ) <-> ( T. /\ ( ps /\ ch ) ) )
4		truan 1501	. . . . 5 |- ( ( T. /\ ( ps /\ ch ) ) <-> ( ps /\ ch ) )
5	3, 4	bitri 264	. . . 4 |- ( ( T. /\ ps /\ ch ) <-> ( ps /\ ch ) )
6		eelT00.1	. . . . 5 |- ( T. -> ph )
7		eelT00.4	. . . . 5 |- ( ( ph /\ ps /\ ch ) -> th )
8	6, 7	syl3an1 1359	. . . 4 |- ( ( T. /\ ps /\ ch ) -> th )
9	5, 8	sylbir 225	. . 3 |- ( ( ps /\ ch ) -> th )
10	2, 9	mpan 706	. 2 |- ( ch -> th )
11	1, 10	ax-mp 5	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)