sbco2w

Theorem.

Arguments:

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

Distinct variable conditions:

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

Hypotheses:

(y bound in phi)

Assertions:

([z/y][y/x]phileftrightarrow[z/x]phi)

Proof:

Hyp Ref Line Expr
Hypo1(y bound in phi)
1dfsbf2([z/x]phileftrightarrowexistsy((yeqz)wedgeexistsx((xeqy)wedgephi)))
dfsb-dvex3([z/y][y/x]phileftrightarrowexistsy((yeqz)wedge[y/x]phi))
dfsb-dvex4([y/x]phileftrightarrowexistsx((xeqy)wedgephi))
4bi.bldan>5(((yeqz)wedge[y/x]phi)leftrightarrow((yeqz)wedgeexistsx((xeqy)wedgephi)))
5eqt-∃-i6(existsy((yeqz)wedge[y/x]phi)leftrightarrowexistsy((yeqz)wedgeexistsx((xeqy)wedgephi)))
3, 6bitr7([z/y][y/x]phileftrightarrowexistsy((yeqz)wedgeexistsx((xeqy)wedgephi)))
2, 7><bitr8([z/y][y/x]phileftrightarrow[z/x]phi)