Theorem e111 38899

Description: A virtual deduction elimination rule (see syl3c 66). (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
e111	|- (. ph ->. ta ).

Proof of Theorem e111

Step	Hyp	Ref	Expression
1		e111.3	. . . . 5 |- (. ph ->. th ).
2	1	in1 38787	. . . 4 |- ( ph -> th )
3		e111.1	. . . . . . 7 |- (. ph ->. ps ).
4	3	in1 38787	. . . . . 6 |- ( ph -> ps )
5		e111.2	. . . . . . 7 |- (. ph ->. ch ).
6	5	in1 38787	. . . . . 6 |- ( ph -> ch )
7		e111.4	. . . . . 6 |- ( ps -> ( ch -> ( th -> ta ) ) )
8	4, 6, 7	syl2im 40	. . . . 5 |- ( ph -> ( ph -> ( th -> ta ) ) )
9	8	pm2.43i 52	. . . 4 |- ( ph -> ( th -> ta ) )
10	2, 9	syl5com 31	. . 3 |- ( ph -> ( ph -> ta ) )
11	10	pm2.43i 52	. 2 |- ( ph -> ta )
12	11	dfvd1ir 38789	1 |- (. ph ->. ta ).

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:  e110  38901  e101  38903  e011  38905  e100  38907  e010  38909  e001  38911  e11  38913  sbcoreleleqVD  39095  ordelordALTVD  39103