# Metamath Proof Explorer

## Theorem bitr3VD

Description: Virtual deduction proof of bitr3 . 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 ) ). 2:1,?: e1a |- (. ( ph <-> ps ) ->. ( ps <-> ph ) ). 3:: |- (. ( ph <-> ps ) ,. ( ph <-> ch ) ->. ( ph <-> ch ) ). 4:3,?: e2 |- (. ( ph <-> ps ) ,. ( ph <-> ch ) ->. ( ch <-> ph ) ). 5:2,4,?: e12 |- (. ( ph <-> ps ) ,. ( ph <-> ch ) ->. ( ps <-> ch ) ). 6:5: |- (. ( ph <-> ps ) ->. ( ( ph <-> ch ) -> ( ps <-> ch ) ) ). qed:6: |- ( ( ph <-> ps ) -> ( ( ph <-> ch ) -> ( ps <-> ch ) ) )
(Contributed by Alan Sare, 31-Dec-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion bitr3VD ${⊢}\left({\phi }↔{\psi }\right)\to \left(\left({\phi }↔{\chi }\right)\to \left({\psi }↔{\chi }\right)\right)$

### Proof

Step Hyp Ref Expression
1 id ${⊢}\left({\phi }↔{\psi }\right)\to \left({\phi }↔{\psi }\right)$
2 1 bicomd ${⊢}\left({\phi }↔{\psi }\right)\to \left({\psi }↔{\phi }\right)$
3 id ${⊢}\left({\phi }↔{\chi }\right)\to \left({\phi }↔{\chi }\right)$
4 3 bicomd ${⊢}\left({\phi }↔{\chi }\right)\to \left({\chi }↔{\phi }\right)$
5 biantr ${⊢}\left(\left({\psi }↔{\phi }\right)\wedge \left({\chi }↔{\phi }\right)\right)\to \left({\psi }↔{\chi }\right)$
6 5 ex ${⊢}\left({\psi }↔{\phi }\right)\to \left(\left({\chi }↔{\phi }\right)\to \left({\psi }↔{\chi }\right)\right)$
7 2 4 6 syl2im ${⊢}\left({\phi }↔{\psi }\right)\to \left(\left({\phi }↔{\chi }\right)\to \left({\psi }↔{\chi }\right)\right)$