bi.bldsb

Theorem.

Arguments:

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

Dummy variables:

z (sv),

Distinct variable conditions:

(z, y), (z, x), (z, phi), (z, psi),

Hypotheses:

(phileftrightarrowpsi)

Assertions:

([y/x]phileftrightarrow[y/x]psi)

Proof:

Hyp Ref Line Expr
Hypo1(phileftrightarrowpsi)
1bi.bldan>2(((xeqz)wedgephi)leftrightarrow((xeqz)wedgepsi))
2eqt-∃-i3(existsx((xeqz)wedgephi)leftrightarrowexistsx((xeqz)wedgepsi))
3bi.bldan>4(((zeqy)wedgeexistsx((xeqz)wedgephi))leftrightarrow((zeqy)wedgeexistsx((xeqz)wedgepsi)))
4eqt-∃-i5(existsz((zeqy)wedgeexistsx((xeqz)wedgephi))leftrightarrowexistsz((zeqy)wedgeexistsx((xeqz)wedgepsi)))
df-sb6([y/x]phileftrightarrowexistsz((zeqy)wedgeexistsx((xeqz)wedgephi)))
df-sb7([y/x]psileftrightarrowexistsz((zeqy)wedgeexistsx((xeqz)wedgepsi)))
5, 6, 7<2bitr8([y/x]phileftrightarrow[y/x]psi)