Theorem eel12131 38938

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

Hypotheses

Ref	Expression
eel12131.1	|- ( ph -> ps )
eel12131.2	|- ( ( ph /\ ch ) -> th )
eel12131.3	|- ( ( ph /\ ta ) -> et )
eel12131.4	|- ( ( ps /\ th /\ et ) -> ze )

Assertion

Ref	Expression
eel12131	|- ( ( ph /\ ch /\ ta ) -> ze )

Proof of Theorem eel12131

Step	Hyp	Ref	Expression
1		eel12131.3	. . . . 5 |- ( ( ph /\ ta ) -> et )
2		eel12131.1	. . . . . . . . 9 |- ( ph -> ps )
3		eel12131.2	. . . . . . . . 9 |- ( ( ph /\ ch ) -> th )
4		eel12131.4	. . . . . . . . . 10 |- ( ( ps /\ th /\ et ) -> ze )
5	4	3exp 1264	. . . . . . . . 9 |- ( ps -> ( th -> ( et -> ze ) ) )
6	2, 3, 5	syl2imc 41	. . . . . . . 8 |- ( ( ph /\ ch ) -> ( ph -> ( et -> ze ) ) )
7	6	ex 450	. . . . . . 7 |- ( ph -> ( ch -> ( ph -> ( et -> ze ) ) ) )
8	7	pm2.43b 55	. . . . . 6 |- ( ch -> ( ph -> ( et -> ze ) ) )
9	8	com13 88	. . . . 5 |- ( et -> ( ph -> ( ch -> ze ) ) )
10	1, 9	syl 17	. . . 4 |- ( ( ph /\ ta ) -> ( ph -> ( ch -> ze ) ) )
11	10	ex 450	. . 3 |- ( ph -> ( ta -> ( ph -> ( ch -> ze ) ) ) )
12	11	pm2.43b 55	. 2 |- ( ta -> ( ph -> ( ch -> ze ) ) )
13	12	3imp231 1258	1 |- ( ( ph /\ ch /\ ta ) -> ze )

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:  isosctrlem1ALT  39170