Theorem eel11111 38950

Description: Five-hypothesis elimination deduction for an assertion with a singleton virtual hypothesis collection. Similar to syl113anc 1338 except the unification theorem uses left-nested conjunction. (Contributed by Alan Sare, 17-Oct-2017.)

Hypotheses

Ref	Expression
eel11111.1	|- ( ph -> ps )
eel11111.2	|- ( ph -> ch )
eel11111.3	|- ( ph -> th )
eel11111.4	|- ( ph -> ta )
eel11111.5	|- ( ph -> et )
eel11111.6	|- ( ( ( ( ( ps /\ ch ) /\ th ) /\ ta ) /\ et ) -> ze )

Assertion

Ref	Expression
eel11111	|- ( ph -> ze )

Proof of Theorem eel11111

Step	Hyp	Ref	Expression
1		eel11111.4	. 2 |- ( ph -> ta )
2		eel11111.5	. 2 |- ( ph -> et )
3		eel11111.1	. . 3 |- ( ph -> ps )
4		eel11111.2	. . 3 |- ( ph -> ch )
5		eel11111.3	. . 3 |- ( ph -> th )
6		eel11111.6	. . . . 5 |- ( ( ( ( ( ps /\ ch ) /\ th ) /\ ta ) /\ et ) -> ze )
7	6	exp41 638	. . . 4 |- ( ( ps /\ ch ) -> ( th -> ( ta -> ( et -> ze ) ) ) )
8	7	ex 450	. . 3 |- ( ps -> ( ch -> ( th -> ( ta -> ( et -> ze ) ) ) ) )
9	3, 4, 5, 8	syl3c 66	. 2 |- ( ph -> ( ta -> ( et -> ze ) ) )
10	1, 2, 9	mp2d 49	1 |- ( ph -> ze )

Syntax hints:   -> wi 4   /\ wa 384

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

This theorem is referenced by: (None)