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Metamath Proof Explorer

Theorem e222

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

Ref Expression
Hypotheses e222.1 
|- (. ph ,. ps ->. ch ).
e222.2 
|- (. ph ,. ps ->. th ).
e222.3 
|- (. ph ,. ps ->. ta ).
e222.4 
|- ( ch -> ( th -> ( ta -> et ) ) )
Assertion e222 
|- (. ph ,. ps ->. et ).

Proof

Step Hyp Ref Expression
1 e222.1 
 |-  (. ph ,. ps ->. ch ).
2 e222.2 
 |-  (. ph ,. ps ->. th ).
3 e222.3 
 |-  (. ph ,. ps ->. ta ).
4 e222.4 
 |-  ( ch -> ( th -> ( ta -> et ) ) )
5 3 dfvd2i 
 |-  ( ph -> ( ps -> ta ) )
6 5 imp 
 |-  ( ( ph /\ ps ) -> ta )
7 1 dfvd2i 
 |-  ( ph -> ( ps -> ch ) )
8 7 imp 
 |-  ( ( ph /\ ps ) -> ch )
9 2 dfvd2i 
 |-  ( ph -> ( ps -> th ) )
10 9 imp 
 |-  ( ( ph /\ ps ) -> th )
11 8 10 4 syl2im 
 |-  ( ( ph /\ ps ) -> ( ( ph /\ ps ) -> ( ta -> et ) ) )
12 11 pm2.43i 
 |-  ( ( ph /\ ps ) -> ( ta -> et ) )
13 6 12 syl5com 
 |-  ( ( ph /\ ps ) -> ( ( ph /\ ps ) -> et ) )
14 13 pm2.43i 
 |-  ( ( ph /\ ps ) -> et )
15 14 ex 
 |-  ( ph -> ( ps -> et ) )
16 15 dfvd2ir 
 |-  (. ph ,. ps ->. et ).