Theorem eelT12 38934

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

Hypotheses

Ref	Expression
eelT12.1	|- ( T. -> ph )
eelT12.2	|- ( ps -> ch )
eelT12.3	|- ( th -> ta )
eelT12.4	|- ( ( ph /\ ch /\ ta ) -> et )

Assertion

Ref	Expression
eelT12	|- ( ( ps /\ th ) -> et )

Proof of Theorem eelT12

Step	Hyp	Ref	Expression
1		3anass 1042	. . 3 |- ( ( T. /\ ps /\ th ) <-> ( T. /\ ( ps /\ th ) ) )
2		truan 1501	. . 3 |- ( ( T. /\ ( ps /\ th ) ) <-> ( ps /\ th ) )
3	1, 2	bitri 264	. 2 |- ( ( T. /\ ps /\ th ) <-> ( ps /\ th ) )
4		eelT12.3	. . 3 |- ( th -> ta )
5		eelT12.2	. . . 4 |- ( ps -> ch )
6		eelT12.1	. . . . 5 |- ( T. -> ph )
7		eelT12.4	. . . . 5 |- ( ( ph /\ ch /\ ta ) -> et )
8	6, 7	syl3an1 1359	. . . 4 |- ( ( T. /\ ch /\ ta ) -> et )
9	5, 8	syl3an2 1360	. . 3 |- ( ( T. /\ ps /\ ta ) -> et )
10	4, 9	syl3an3 1361	. 2 |- ( ( T. /\ ps /\ th ) -> et )
11	3, 10	sylbir 225	1 |- ( ( ps /\ th ) -> et )

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)