Theorem e223

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

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

Proof

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