eqsal

Theorem.

Arguments:

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

Distinct variable conditions:

(x, y),

Hypotheses:

((xeqy)to(phileftrightarrowpsi))
(x bound in psi)

Assertions:

(forallx((xeqy)tophi)leftrightarrowpsi)

Proof:

Hyp Ref Line Expr
Hypo1((xeqy)to(phileftrightarrowpsi))
2(x bound in psi)
2fv.eqal3(psileftrightarrowforallxpsi)
1, 3rpb33>>4((xeqy)to(phileftrightarrowforallxpsi))
4im.bidi5(((xeqy)tophi)leftrightarrow((xeqy)toforallxpsi))
5eqt-∀-i6(forallx((xeqy)tophi)leftrightarrowforallx((xeqy)toforallxpsi))
ax17(forallxpsito((xeqy)toforallxpsi))
7ax58(forallxpsitoforallx((xeqy)toforallxpsi))
2df-Bvp9(psitoforallxpsi)
8, 9syl10(psitoforallx((xeqy)toforallxpsi))
ax9alt11(forallx((xeqy)toforallxpsi)topsi)
10, 11>bii12(psileftrightarrowforallx((xeqy)toforallxpsi))
6, 12><bitr13(forallx((xeqy)tophi)leftrightarrowpsi)