Theorem eel0T1 38937

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

Hypotheses

Ref	Expression
eel0T1.1	|- ph
eel0T1.2	|- ( T. -> ps )
eel0T1.3	|- ( ch -> th )
eel0T1.4	|- ( ( ph /\ ps /\ th ) -> ta )

Assertion

Ref	Expression
eel0T1	|- ( ch -> ta )

Proof of Theorem eel0T1

Step	Hyp	Ref	Expression
1		3anass 1042	. . 3 |- ( ( ph /\ T. /\ ch ) <-> ( ph /\ ( T. /\ ch ) ) )
2		simpr 477	. . . 4 |- ( ( ph /\ ( T. /\ ch ) ) -> ( T. /\ ch ) )
3		eel0T1.1	. . . . 5 |- ph
4	3	jctl 564	. . . 4 |- ( ( T. /\ ch ) -> ( ph /\ ( T. /\ ch ) ) )
5	2, 4	impbii 199	. . 3 |- ( ( ph /\ ( T. /\ ch ) ) <-> ( T. /\ ch ) )
6		truan 1501	. . 3 |- ( ( T. /\ ch ) <-> ch )
7	1, 5, 6	3bitri 286	. 2 |- ( ( ph /\ T. /\ ch ) <-> ch )
8		eel0T1.3	. . 3 |- ( ch -> th )
9		eel0T1.2	. . . 4 |- ( T. -> ps )
10		eel0T1.4	. . . 4 |- ( ( ph /\ ps /\ th ) -> ta )
11	9, 10	syl3an2 1360	. . 3 |- ( ( ph /\ T. /\ th ) -> ta )
12	8, 11	syl3an3 1361	. 2 |- ( ( ph /\ T. /\ ch ) -> ta )
13	7, 12	sylbir 225	1 |- ( ch -> ta )

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)