sbal-w

Theorem.

Arguments:

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

Distinct variable conditions:

(z, a), (z, x), (x, a),

Assertions:

([a/x]forallzphileftrightarrowforallz[a/x]phi)

Proof:

Hyp Ref Line Expr
ax17-bv1(z bound in (xeqa))
1imgen>2(forallz((xeqa)tophi)leftrightarrow((xeqa)toforallzphi))
2eqt-∀-i3(forallxforallz((xeqa)tophi)leftrightarrowforallx((xeqa)toforallzphi))
alcom4(forallzforallx((xeqa)tophi)leftrightarrowforallxforallz((xeqa)tophi))
3, 4bitr5(forallzforallx((xeqa)tophi)leftrightarrowforallx((xeqa)toforallzphi))
dfsb-dval6([a/x]forallzphileftrightarrowforallx((xeqa)toforallzphi))
dfsb-dval7([a/x]phileftrightarrowforallx((xeqa)tophi))
7eqt-∀-i8(forallz[a/x]phileftrightarrowforallzforallx((xeqa)tophi))
5, 6, 8<2bitr9(forallz[a/x]phileftrightarrow[a/x]forallzphi)
9bicomi10([a/x]forallzphileftrightarrowforallz[a/x]phi)