Theorem e111

Description: A virtual deduction elimination rule (see syl3c ). (Contributed by Alan Sare, 14-Jun-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	e111.1	|- (. ph ->. ps ).
	e111.2	|- (. ph ->. ch ).
	e111.3	|- (. ph ->. th ).
	e111.4	|- ( ps -> ( ch -> ( th -> ta ) ) )
Assertion	e111	|- (. ph ->. ta ).

Proof

Step	Hyp	Ref	Expression
1		e111.1	|- (. ph ->. ps ).
2		e111.2	|- (. ph ->. ch ).
3		e111.3	|- (. ph ->. th ).
4		e111.4	|- ( ps -> ( ch -> ( th -> ta ) ) )
5	3	in1	|- ( ph -> th )
6	1	in1	|- ( ph -> ps )
7	2	in1	|- ( ph -> ch )
8	6 7 4	syl2im	|- ( ph -> ( ph -> ( th -> ta ) ) )
9	8	pm2.43i	|- ( ph -> ( th -> ta ) )
10	5 9	syl5com	|- ( ph -> ( ph -> ta ) )
11	10	pm2.43i	|- ( ph -> ta )
12	11	dfvd1ir	|- (. ph ->. ta ).