# 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 ${⊢}\left({\phi }\to \left({\psi }\to {\chi }\right)\right)\to \left(\left({\theta }\to {\psi }\right)\to \left({\phi }\to \left({\theta }\to {\chi }\right)\right)\right)$

### Proof

Step Hyp Ref Expression
1 idn2 ${⊢}\left(\left({\phi }\to \left({\psi }\to {\chi }\right)\right){,}\left({\theta }\to {\psi }\right){\to }\left({\theta }\to {\psi }\right)\right)$
2 idn1 ${⊢}\left(\left({\phi }\to \left({\psi }\to {\chi }\right)\right){\to }\left({\phi }\to \left({\psi }\to {\chi }\right)\right)\right)$
3 pm2.04 ${⊢}\left({\phi }\to \left({\psi }\to {\chi }\right)\right)\to \left({\psi }\to \left({\phi }\to {\chi }\right)\right)$
4 2 3 e1a ${⊢}\left(\left({\phi }\to \left({\psi }\to {\chi }\right)\right){\to }\left({\psi }\to \left({\phi }\to {\chi }\right)\right)\right)$
5 imim1 ${⊢}\left({\theta }\to {\psi }\right)\to \left(\left({\psi }\to \left({\phi }\to {\chi }\right)\right)\to \left({\theta }\to \left({\phi }\to {\chi }\right)\right)\right)$
6 1 4 5 e21 ${⊢}\left(\left({\phi }\to \left({\psi }\to {\chi }\right)\right){,}\left({\theta }\to {\psi }\right){\to }\left({\theta }\to \left({\phi }\to {\chi }\right)\right)\right)$
7 pm2.04 ${⊢}\left({\theta }\to \left({\phi }\to {\chi }\right)\right)\to \left({\phi }\to \left({\theta }\to {\chi }\right)\right)$
8 6 7 e2 ${⊢}\left(\left({\phi }\to \left({\psi }\to {\chi }\right)\right){,}\left({\theta }\to {\psi }\right){\to }\left({\phi }\to \left({\theta }\to {\chi }\right)\right)\right)$
9 8 in2 ${⊢}\left(\left({\phi }\to \left({\psi }\to {\chi }\right)\right){\to }\left(\left({\theta }\to {\psi }\right)\to \left({\phi }\to \left({\theta }\to {\chi }\right)\right)\right)\right)$
10 9 in1 ${⊢}\left({\phi }\to \left({\psi }\to {\chi }\right)\right)\to \left(\left({\theta }\to {\psi }\right)\to \left({\phi }\to \left({\theta }\to {\chi }\right)\right)\right)$