Theorem el2122old 38944

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

Hypotheses

Ref	Expression
el2122old.1	|- (. (. ph ,. ps ). ->. ch ).
el2122old.2	|- (. ps ->. th ).
el2122old.3	|- (. ps ->. ta ).
el2122old.4	|- ( ( ch /\ th /\ ta ) -> et )

Assertion

Ref	Expression
el2122old	|- (. (. ph ,. ps ). ->. et ).

Proof of Theorem el2122old

Step	Hyp	Ref	Expression
1		el2122old.1	. . . 4 |- (. (. ph ,. ps ). ->. ch ).
2	1	dfvd2ani 38799	. . 3 |- ( ( ph /\ ps ) -> ch )
3		el2122old.2	. . . 4 |- (. ps ->. th ).
4	3	in1 38787	. . 3 |- ( ps -> th )
5		el2122old.3	. . . 4 |- (. ps ->. ta ).
6	5	in1 38787	. . 3 |- ( ps -> ta )
7		el2122old.4	. . 3 |- ( ( ch /\ th /\ ta ) -> et )
8	2, 4, 6, 7	eel2122old 38943	. 2 |- ( ( ph /\ ps ) -> et )
9	8	dfvd2anir 38800	1 |- (. (. ph ,. ps ). ->. et ).

Syntax hints:   -> wi 4   /\ w3a 1037   (.wvd1 38785   (.wvhc2 38796

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-vhc2 38797

This theorem is referenced by:  suctrALTcfVD  39159