# Metamath Proof Explorer

## Theorem 3ornot23VD

Description: Virtual deduction proof of 3ornot23 . 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 ) ->. ( -. ph /\ -. ps ) ). 2:: |- (. ( -. ph /\ -. ps ) ,. ( ch \/ ph \/ ps ) ->. ( ch \/ ph \/ ps ) ). 3:1,?: e1a |- (. ( -. ph /\ -. ps ) ->. -. ph ). 4:1,?: e1a |- (. ( -. ph /\ -. ps ) ->. -. ps ). 5:3,4,?: e11 |- (. ( -. ph /\ -. ps ) ->. -. ( ph \/ ps ) ). 6:2,?: e2 |- (. ( -. ph /\ -. ps ) ,. ( ch \/ ph \/ ps ) ->. ( ch \/ ( ph \/ ps ) ) ). 7:5,6,?: e12 |- (. ( -. ph /\ -. ps ) ,. ( ch \/ ph \/ ps ) ->. ch ). 8:7: |- (. ( -. ph /\ -. ps ) ->. ( ( ch \/ ph \/ ps ) -> ch ) ). qed:8: |- ( ( -. ph /\ -. ps ) -> ( ( ch \/ ph \/ ps ) -> ch ) )
(Contributed by Alan Sare, 31-Dec-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion 3ornot23VD ${⊢}\left(¬{\phi }\wedge ¬{\psi }\right)\to \left(\left({\chi }\vee {\phi }\vee {\psi }\right)\to {\chi }\right)$

### Proof

Step Hyp Ref Expression
1 idn1 ${⊢}\left(\left(¬{\phi }\wedge ¬{\psi }\right){\to }\left(¬{\phi }\wedge ¬{\psi }\right)\right)$
2 simpl ${⊢}\left(¬{\phi }\wedge ¬{\psi }\right)\to ¬{\phi }$
3 1 2 e1a ${⊢}\left(\left(¬{\phi }\wedge ¬{\psi }\right){\to }¬{\phi }\right)$
4 simpr ${⊢}\left(¬{\phi }\wedge ¬{\psi }\right)\to ¬{\psi }$
5 1 4 e1a ${⊢}\left(\left(¬{\phi }\wedge ¬{\psi }\right){\to }¬{\psi }\right)$
6 ioran ${⊢}¬\left({\phi }\vee {\psi }\right)↔\left(¬{\phi }\wedge ¬{\psi }\right)$
7 6 simplbi2 ${⊢}¬{\phi }\to \left(¬{\psi }\to ¬\left({\phi }\vee {\psi }\right)\right)$
8 3 5 7 e11 ${⊢}\left(\left(¬{\phi }\wedge ¬{\psi }\right){\to }¬\left({\phi }\vee {\psi }\right)\right)$
9 idn2 ${⊢}\left(\left(¬{\phi }\wedge ¬{\psi }\right){,}\left({\chi }\vee {\phi }\vee {\psi }\right){\to }\left({\chi }\vee {\phi }\vee {\psi }\right)\right)$
10 3orass ${⊢}\left({\chi }\vee {\phi }\vee {\psi }\right)↔\left({\chi }\vee \left({\phi }\vee {\psi }\right)\right)$
11 10 biimpi ${⊢}\left({\chi }\vee {\phi }\vee {\psi }\right)\to \left({\chi }\vee \left({\phi }\vee {\psi }\right)\right)$
12 9 11 e2 ${⊢}\left(\left(¬{\phi }\wedge ¬{\psi }\right){,}\left({\chi }\vee {\phi }\vee {\psi }\right){\to }\left({\chi }\vee \left({\phi }\vee {\psi }\right)\right)\right)$
13 orel2 ${⊢}¬\left({\phi }\vee {\psi }\right)\to \left(\left({\chi }\vee \left({\phi }\vee {\psi }\right)\right)\to {\chi }\right)$
14 8 12 13 e12 ${⊢}\left(\left(¬{\phi }\wedge ¬{\psi }\right){,}\left({\chi }\vee {\phi }\vee {\psi }\right){\to }{\chi }\right)$
15 14 in2 ${⊢}\left(\left(¬{\phi }\wedge ¬{\psi }\right){\to }\left(\left({\chi }\vee {\phi }\vee {\psi }\right)\to {\chi }\right)\right)$
16 15 in1 ${⊢}\left(¬{\phi }\wedge ¬{\psi }\right)\to \left(\left({\chi }\vee {\phi }\vee {\psi }\right)\to {\chi }\right)$