Metamath Proof Explorer


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 ).