eqs4lem

Theorem.

Arguments:

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

Distinct variable conditions:

(x, y),

Assertions:

(forallx((xeqy)tophi)toexistsx((xeqy)wedgephi))

Proof:

Hyp Ref Line Expr
ax41(forallx((xeqy)tophi)to((xeqy)tophi))
ax42(forallx((xeqy)tolnotphi)to((xeqy)tolnotphi))
1, 2msca3(forallx((xeqy)tophi)to((xeqy)tolnotforallx((xeqy)tolnotphi)))
fv-nalx4(x bound in lnotforallx((xeqy)tolnotphi))
4df-Bvp5(lnotforallx((xeqy)tolnotphi)toforallxlnotforallx((xeqy)tolnotphi))
3, 5rpi336(forallx((xeqy)tophi)to((xeqy)toforallxlnotforallx((xeqy)tolnotphi)))
6ax57(forallx((xeqy)tophi)toforallx((xeqy)toforallxlnotforallx((xeqy)tolnotphi)))
ax9alt8(forallx((xeqy)toforallxlnotforallx((xeqy)tolnotphi))tolnotforallx((xeqy)tolnotphi))
7, 8syl9(forallx((xeqy)tophi)tolnotforallx((xeqy)tolnotphi))
eqs3lem10(existsx((xeqy)wedgephi)leftrightarrowlnotforallx((xeqy)tolnotphi))
9, 10brpi22<11(forallx((xeqy)tophi)toexistsx((xeqy)wedgephi))

Theorems:

NameComment
msca