Theorem e1111 38900

Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 6-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
e1111.1	|- (. ph ->. ps ).
e1111.2	|- (. ph ->. ch ).
e1111.3	|- (. ph ->. th ).
e1111.4	|- (. ph ->. ta ).
e1111.5	|- ( ps -> ( ch -> ( th -> ( ta -> et ) ) ) )

Assertion

Ref	Expression
e1111	|- (. ph ->. et ).

Proof of Theorem e1111

Step	Hyp	Ref	Expression
1		e1111.1	. . . 4 |- (. ph ->. ps ).
2	1	in1 38787	. . 3 |- ( ph -> ps )
3		e1111.2	. . . 4 |- (. ph ->. ch ).
4	3	in1 38787	. . 3 |- ( ph -> ch )
5		e1111.3	. . . 4 |- (. ph ->. th ).
6	5	in1 38787	. . 3 |- ( ph -> th )
7		e1111.4	. . . 4 |- (. ph ->. ta ).
8	7	in1 38787	. . 3 |- ( ph -> ta )
9		e1111.5	. . 3 |- ( ps -> ( ch -> ( th -> ( ta -> et ) ) ) )
10	2, 4, 6, 8, 9	ee1111 38722	. 2 |- ( ph -> et )
11	10	dfvd1ir 38789	1 |- (. ph ->. et ).

Syntax hints:   -> wi 4   (.wvd1 38785

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-vd1 38786

This theorem is referenced by:  trsbcVD  39113