sb-orw

Theorem.

Arguments:

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

Distinct variable conditions:

(a, x),

Assertions:

([a/x](phiveepsi)leftrightarrow([a/x]phivee[a/x]psi))

Proof:

Hyp Ref Line Expr
dfsb-dvex1([a/x](phiveepsi)leftrightarrowexistsx((xeqa)wedge(phiveepsi)))
an.ordi<2(((xeqa)wedge(phiveepsi))leftrightarrow(((xeqa)wedgephi)vee((xeqa)wedgepsi)))
2eqt-∃-i3(existsx((xeqa)wedge(phiveepsi))leftrightarrowexistsx(((xeqa)wedgephi)vee((xeqa)wedgepsi)))
ex.ordi4(existsx(((xeqa)wedgephi)vee((xeqa)wedgepsi))leftrightarrow(existsx((xeqa)wedgephi)veeexistsx((xeqa)wedgepsi)))
1, 3, 42bitr5([a/x](phiveepsi)leftrightarrow(existsx((xeqa)wedgephi)veeexistsx((xeqa)wedgepsi)))
dfsb-dvex6([a/x]phileftrightarrowexistsx((xeqa)wedgephi))
dfsb-dvex7([a/x]psileftrightarrowexistsx((xeqa)wedgepsi))
6, 7bi.bldor<>8(([a/x]phivee[a/x]psi)leftrightarrow(existsx((xeqa)wedgephi)veeexistsx((xeqa)wedgepsi)))
5, 8><bitr9([a/x](phiveepsi)leftrightarrow([a/x]phivee[a/x]psi))