cbv2-dv

Theorem.

Arguments:

x (sv), y (sv), x (sv), y (sv), phi (pr), psi (pr), chi (pr),

Distinct variable conditions:

(x, y), (z, x), (z, y),

Hypotheses:

(phito((xeqy)to(psileftrightarrowchi)))
(phito(chitoforallxchi))
(phito(psitoforallypsi))

Assertions:

(forallxforallyphito(forallxpsileftrightarrowforallychi))

Proof:

Hyp Ref Line Expr
Hypo1(phito((xeqy)to(psileftrightarrowchi)))
2(phito(chitoforallxchi))
3(phito(psitoforallypsi))
bi>4((psileftrightarrowchi)to(psitochi))
1, 4rpi335(phito((xeqy)to(psitochi)))
2, 3, 5cbv1-dv6(forallxforallyphito(forallxpsitoforallychi))
bi<7((psileftrightarrowchi)to(chitopsi))
1, 7rpi338(phito((xeqy)to(chitopsi)))
eqcomi9((yeqx)to(xeqy))
8, 9rpi3210(phito((yeqx)to(chitopsi)))
2, 3, 10cbv1-dv11(forallyforallxphito(forallychitoforallxpsi))
ax712(forallxforallyphitoforallyforallxphi)
11, 12syl13(forallxforallyphito(forallychitoforallxpsi))
6, 13>bid14(forallxforallyphito(forallxpsileftrightarrowforallychi))