Metamath Proof Explorer

Theorem syl5impVD

Description: Virtual deduction proof of syl5imp . The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.

1:: |- (. ( ph -> ( ps -> ch ) ) ->. ( ph -> ( ps -> ch ) ) ).
2:1,?: e1a |- (. ( ph -> ( ps -> ch ) ) ->. ( ps -> ( ph -> ch ) ) ).
3:: |- (. ( ph -> ( ps -> ch ) ) ,. ( th -> ps ) ->. ( th -> ps ) ).
4:3,2,?: e21 |- (. ( ph -> ( ps -> ch ) ) ,. ( th -> ps ) ->. ( th -> ( ph -> ch ) ) ).
5:4,?: e2 |- (. ( ph -> ( ps -> ch ) ) ,. ( th -> ps ) ->. ( ph -> ( th -> ch ) ) ).
6:5: |- (. ( ph -> ( ps -> ch ) ) ->. ( ( th -> ps ) -> ( ph -> ( th -> ch ) ) ) ).
qed:6: |- ( ( ph -> ( ps -> ch ) ) -> ( ( th -> ps ) -> ( ph -> ( th -> ch ) ) ) )
(Contributed by Alan Sare, 31-Dec-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion syl5impVD 
|- ( ( ph -> ( ps -> ch ) ) -> ( ( th -> ps ) -> ( ph -> ( th -> ch ) ) ) )

Proof

Step Hyp Ref Expression
1 idn2 
 |-  (. ( ph -> ( ps -> ch ) ) ,. ( th -> ps ) ->. ( th -> ps ) ).
2 idn1 
 |-  (. ( ph -> ( ps -> ch ) ) ->. ( ph -> ( ps -> ch ) ) ).
3 pm2.04 
 |-  ( ( ph -> ( ps -> ch ) ) -> ( ps -> ( ph -> ch ) ) )
4 2 3 e1a 
 |-  (. ( ph -> ( ps -> ch ) ) ->. ( ps -> ( ph -> ch ) ) ).
5 imim1 
 |-  ( ( th -> ps ) -> ( ( ps -> ( ph -> ch ) ) -> ( th -> ( ph -> ch ) ) ) )
6 1 4 5 e21 
 |-  (. ( ph -> ( ps -> ch ) ) ,. ( th -> ps ) ->. ( th -> ( ph -> ch ) ) ).
7 pm2.04 
 |-  ( ( th -> ( ph -> ch ) ) -> ( ph -> ( th -> ch ) ) )
8 6 7 e2 
 |-  (. ( ph -> ( ps -> ch ) ) ,. ( th -> ps ) ->. ( ph -> ( th -> ch ) ) ).
9 8 in2 
 |-  (. ( ph -> ( ps -> ch ) ) ->. ( ( th -> ps ) -> ( ph -> ( th -> ch ) ) ) ).
10 9 in1 
 |-  ( ( ph -> ( ps -> ch ) ) -> ( ( th -> ps ) -> ( ph -> ( th -> ch ) ) ) )