Theorem e223 31659

Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
e223.1	|- (. ph ,. ps ->. ch ).
e223.2	|- (. ph ,. ps ->. th ).
e223.3	|- (. ph ,. ps ,. ta ->. et ).
e223.4	|- ( ch -> ( th -> ( et -> ze ) ) )

Assertion

Ref	Expression
e223	|- (. ph ,. ps ,. ta ->. ze ).

Proof of Theorem e223

Step	Hyp	Ref	Expression
1		e223.1	. . . . 5 |- (. ph ,. ps ->. ch ).
2	1	in2 31629	. . . 4 |- (. ph ->. ( ps -> ch ) ).
3	2	in1 31586	. . 3 |- ( ph -> ( ps -> ch ) )
4		e223.2	. . . . 5 |- (. ph ,. ps ->. th ).
5	4	in2 31629	. . . 4 |- (. ph ->. ( ps -> th ) ).
6	5	in1 31586	. . 3 |- ( ph -> ( ps -> th ) )
7		e223.3	. . . . . 6 |- (. ph ,. ps ,. ta ->. et ).
8	7	in3 31633	. . . . 5 |- (. ph ,. ps ->. ( ta -> et ) ).
9	8	in2 31629	. . . 4 |- (. ph ->. ( ps -> ( ta -> et ) ) ).
10	9	in1 31586	. . 3 |- ( ph -> ( ps -> ( ta -> et ) ) )
11		e223.4	. . 3 |- ( ch -> ( th -> ( et -> ze ) ) )
12	3, 6, 10, 11	ee223 31658	. 2 |- ( ph -> ( ps -> ( ta -> ze ) ) )
13	12	dfvd3ir 31608	1 |- (. ph ,. ps ,. ta ->. ze ).

Syntax hints:   -> wi 4   (.wvd2 31592   (.wvd3 31602

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 185  df-an 371  df-3an 967  df-vd1 31585  df-vd2 31593  df-vd3 31605

This theorem is referenced by:  tratrbVD  31899