Theorem 3imp3i2an 1278

Description: An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.)

Hypotheses

Ref	Expression
3imp3i2an.1	|- ( ( ph /\ ps /\ ch ) -> th )
3imp3i2an.2	|- ( ( ph /\ ch ) -> ta )
3imp3i2an.3	|- ( ( th /\ ta ) -> et )

Assertion

Ref	Expression
3imp3i2an	|- ( ( ph /\ ps /\ ch ) -> et )

Proof of Theorem 3imp3i2an

Step	Hyp	Ref	Expression
1		3imp3i2an.2	. . . . . . 7 |- ( ( ph /\ ch ) -> ta )
2		3imp3i2an.1	. . . . . . . . . 10 |- ( ( ph /\ ps /\ ch ) -> th )
3	2	3exp 1264	. . . . . . . . 9 |- ( ph -> ( ps -> ( ch -> th ) ) )
4		3imp3i2an.3	. . . . . . . . . 10 |- ( ( th /\ ta ) -> et )
5	4	ex 450	. . . . . . . . 9 |- ( th -> ( ta -> et ) )
6	3, 5	syl8 76	. . . . . . . 8 |- ( ph -> ( ps -> ( ch -> ( ta -> et ) ) ) )
7	6	com4r 94	. . . . . . 7 |- ( ta -> ( ph -> ( ps -> ( ch -> et ) ) ) )
8	1, 7	syl 17	. . . . . 6 |- ( ( ph /\ ch ) -> ( ph -> ( ps -> ( ch -> et ) ) ) )
9	8	ex 450	. . . . 5 |- ( ph -> ( ch -> ( ph -> ( ps -> ( ch -> et ) ) ) ) )
10	9	pm2.43b 55	. . . 4 |- ( ch -> ( ph -> ( ps -> ( ch -> et ) ) ) )
11	10	com4r 94	. . 3 |- ( ch -> ( ch -> ( ph -> ( ps -> et ) ) ) )
12	11	pm2.43i 52	. 2 |- ( ch -> ( ph -> ( ps -> et ) ) )
13	12	3imp231 1258	1 |- ( ( ph /\ ps /\ ch ) -> et )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037

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

This theorem is referenced by:  upgr2pthnlp  26628  frgrreg  27252  eliuniin  39279  eliuniin2  39303