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Metamath Proof Explorer

Theorem e333

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

Ref Expression
Hypotheses e333.1 
|- (. ph ,. ps ,. ch ->. th ).
e333.2 
|- (. ph ,. ps ,. ch ->. ta ).
e333.3 
|- (. ph ,. ps ,. ch ->. et ).
e333.4 
|- ( th -> ( ta -> ( et -> ze ) ) )
Assertion e333 
|- (. ph ,. ps ,. ch ->. ze ).

Proof

Step Hyp Ref Expression
1 e333.1 
 |-  (. ph ,. ps ,. ch ->. th ).
2 e333.2 
 |-  (. ph ,. ps ,. ch ->. ta ).
3 e333.3 
 |-  (. ph ,. ps ,. ch ->. et ).
4 e333.4 
 |-  ( th -> ( ta -> ( et -> ze ) ) )
5 3 dfvd3i 
 |-  ( ph -> ( ps -> ( ch -> et ) ) )
6 5 3imp 
 |-  ( ( ph /\ ps /\ ch ) -> et )
7 1 dfvd3i 
 |-  ( ph -> ( ps -> ( ch -> th ) ) )
8 7 3imp 
 |-  ( ( ph /\ ps /\ ch ) -> th )
9 2 dfvd3i 
 |-  ( ph -> ( ps -> ( ch -> ta ) ) )
10 9 3imp 
 |-  ( ( ph /\ ps /\ ch ) -> ta )
11 8 10 4 syl2im 
 |-  ( ( ph /\ ps /\ ch ) -> ( ( ph /\ ps /\ ch ) -> ( et -> ze ) ) )
12 11 pm2.43i 
 |-  ( ( ph /\ ps /\ ch ) -> ( et -> ze ) )
13 6 12 syl5com 
 |-  ( ( ph /\ ps /\ ch ) -> ( ( ph /\ ps /\ ch ) -> ze ) )
14 13 pm2.43i 
 |-  ( ( ph /\ ps /\ ch ) -> ze )
15 14 3exp 
 |-  ( ph -> ( ps -> ( ch -> ze ) ) )
16 15 dfvd3ir 
 |-  (. ph ,. ps ,. ch ->. ze ).