df-∃1-alt5

Theorem.

Arguments:

x (sv), phi (pr),

Dummy variables:

y (sv),

Distinct variable conditions:

(y, x), (y, phi),

Assertions:

(_e1xphileftrightarrow(existsxphiwedge_em1xphi))

Proof:

Hyp Ref Line Expr
ax17-bv1(y bound in phi)
1df-∃1-alt32(_e1xphileftrightarrow(existsxphiwedgeexistsyforallx(phito(xeqy))))
1df-∃mo-alt23(_em1xphileftrightarrowexistsyforallx(phito(xeqy)))
3bi.bldan>4((existsxphiwedge_em1xphi)leftrightarrow(existsxphiwedgeexistsyforallx(phito(xeqy))))
2, 4><bitr5(_e1xphileftrightarrow(existsxphiwedge_em1xphi))