df-∃1-alt3

Theorem.

Arguments:

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

Distinct variable conditions:

(x, y),

Hypotheses:

(y bound in phi)

Assertions:

(_e1xphileftrightarrow(existsxphiwedgeexistsyforallx(phito(xeqy))))

Proof:

Hyp Ref Line Expr
Hypo1(y bound in phi)
1df-∃1-alt22(_e1xphileftrightarrow(existsxphiwedgeforallxforally((phiwedge[y/x]phi)to(xeqy))))
1df-∃mo-alt03(existsyforallx(phito(xeqy))leftrightarrowforallxforally((phiwedge[y/x]phi)to(xeqy)))
3bi.bldan>4((existsxphiwedgeexistsyforallx(phito(xeqy)))leftrightarrow(existsxphiwedgeforallxforally((phiwedge[y/x]phi)to(xeqy))))
2, 4><bitr5(_e1xphileftrightarrow(existsxphiwedgeexistsyforallx(phito(xeqy))))