# Metamath Proof Explorer

## Theorem 3impexpbicomiVD

Description: Virtual deduction proof of 3impexpbicomi . 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.

 h1:: |- ( ( ph /\ ps /\ ch ) -> ( th <-> ta ) ) qed:1,?: e0a |- ( ph -> ( ps -> ( ch -> ( ta <-> th ) ) ) )
(Contributed by Alan Sare, 31-Dec-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypothesis 3impexpbicomiVD.1 ${⊢}\left({\phi }\wedge {\psi }\wedge {\chi }\right)\to \left({\theta }↔{\tau }\right)$
Assertion 3impexpbicomiVD ${⊢}{\phi }\to \left({\psi }\to \left({\chi }\to \left({\tau }↔{\theta }\right)\right)\right)$

### Proof

Step Hyp Ref Expression
1 3impexpbicomiVD.1 ${⊢}\left({\phi }\wedge {\psi }\wedge {\chi }\right)\to \left({\theta }↔{\tau }\right)$
2 3impexpbicom ${⊢}\left(\left({\phi }\wedge {\psi }\wedge {\chi }\right)\to \left({\theta }↔{\tau }\right)\right)↔\left({\phi }\to \left({\psi }\to \left({\chi }\to \left({\tau }↔{\theta }\right)\right)\right)\right)$
3 2 biimpi ${⊢}\left(\left({\phi }\wedge {\psi }\wedge {\chi }\right)\to \left({\theta }↔{\tau }\right)\right)\to \left({\phi }\to \left({\psi }\to \left({\chi }\to \left({\tau }↔{\theta }\right)\right)\right)\right)$
4 1 3 e0a ${⊢}{\phi }\to \left({\psi }\to \left({\chi }\to \left({\tau }↔{\theta }\right)\right)\right)$