im.bidi

Theorem.

Arguments:

phi (pr), psi (pr), chi (pr),

Assertions:

((phito(psileftrightarrowchi))leftrightarrow((phitopsi)leftrightarrow(phitochi)))

Proof:

Hyp Ref Line Expr
bi>1((psileftrightarrowchi)to(psitochi))
1syl<2((phito(psileftrightarrowchi))to(phito(psitochi)))
2ax23((phito(psileftrightarrowchi))to((phitopsi)to(phitochi)))
bi<4((psileftrightarrowchi)to(chitopsi))
4syl<5((phito(psileftrightarrowchi))to(phito(chitopsi)))
5ax26((phito(psileftrightarrowchi))to((phitochi)to(phitopsi)))
3, 6>bid7((phito(psileftrightarrowchi))to((phitopsi)leftrightarrow(phitochi)))
bi>8(((phitopsi)leftrightarrow(phitochi))to((phitopsi)to(phitochi)))
8ax2-converse9(((phitopsi)leftrightarrow(phitochi))to(phito(psitochi)))
bi<10(((phitopsi)leftrightarrow(phitochi))to((phitochi)to(phitopsi)))
10ax2-converse11(((phitopsi)leftrightarrow(phitochi))to(phito(chitopsi)))
9, 11im.bldan<>d12(((phitopsi)leftrightarrow(phitochi))to((phiwedgephi)to((psitochi)wedge(chitopsi))))
anidm13((phiwedgephi)leftrightarrowphi)
13bicomi14(phileftrightarrow(phiwedgephi))
dfbi15((psileftrightarrowchi)leftrightarrow((psitochi)wedge(chitopsi)))
12, 14, 15<imtrg16(((phitopsi)leftrightarrow(phitochi))to(phito(psileftrightarrowchi)))
7, 16>bii17((phito(psileftrightarrowchi))leftrightarrow((phitopsi)leftrightarrow(phitochi)))