Theorem el123 38991

Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
el123.1	|- (. ph ->. ps ).
el123.2	|- (. ch ->. th ).
el123.3	|- (. ta ->. et ).
el123.4	|- ( ( ps /\ th /\ et ) -> ze )

Assertion

Ref	Expression
el123	|- (. (. ph ,. ch ,. ta ). ->. ze ).

Proof of Theorem el123

Step	Hyp	Ref	Expression
1		el123.1	. . . 4 |- (. ph ->. ps ).
2	1	in1 38787	. . 3 |- ( ph -> ps )
3		el123.2	. . . 4 |- (. ch ->. th ).
4	3	in1 38787	. . 3 |- ( ch -> th )
5		el123.3	. . . 4 |- (. ta ->. et ).
6	5	in1 38787	. . 3 |- ( ta -> et )
7		el123.4	. . 3 |- ( ( ps /\ th /\ et ) -> ze )
8	2, 4, 6, 7	syl3an 1368	. 2 |- ( ( ph /\ ch /\ ta ) -> ze )
9	8	dfvd3anir 38812	1 |- (. (. ph ,. ch ,. ta ). ->. ze ).

Syntax hints:   -> wi 4   /\ w3a 1037   (.wvd1 38785   (.wvhc3 38804

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-vd1 38786  df-vhc3 38805

This theorem is referenced by:  suctrALTcfVD  39159