chvar-∀-1w

Theorem.

Arguments:

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

Distinct variable conditions:

(y, x),

Assertions:

(forally[y/x]phileftrightarrowforallx[x/y]phi)

Proof:

Hyp Ref Line Expr
dfsb-dval1([y/x]phileftrightarrowforallx((xeqy)tophi))
1eqt-∀-i2(forally[y/x]phileftrightarrowforallyforallx((xeqy)tophi))
alcom3(forallyforallx((xeqy)tophi)leftrightarrowforallxforally((xeqy)tophi))
com4((xeqy)leftrightarrow(yeqx))
4bi.bldim<5(((xeqy)tophi)leftrightarrow((yeqx)tophi))
5eqt-∀-i6(forally((xeqy)tophi)leftrightarrowforally((yeqx)tophi))
dfsb-dval7([x/y]phileftrightarrowforally((yeqx)tophi))
6, 7><bitr8(forally((xeqy)tophi)leftrightarrow[x/y]phi)
8eqt-∀-i9(forallxforally((xeqy)tophi)leftrightarrowforallx[x/y]phi)
2, 3, 92bitr10(forally[y/x]phileftrightarrowforallx[x/y]phi)