df-∃mo-alt2

Theorem.

Arguments:

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

Distinct variable conditions:

(x, y),

Hypotheses:

(y bound in phi)

Assertions:

(_em1xphileftrightarrowexistsyforallx(phito(xeqy)))

Proof:

Hyp Ref Line Expr
Hypo1(y bound in phi)
df-∃mo2(_em1xphileftrightarrow(existsxphito_e1xphi))
aln.ex3(forallxlnotphileftrightarrowlnotexistsxphi)
<neg4(lnotphito(phito(xeqy)))
4im.bldal5(forallxlnotphitoforallx(phito(xeqy)))
spec-ex6(forallx(phito(xeqy))toexistsyforallx(phito(xeqy)))
5, 6syl7(forallxlnotphitoexistsyforallx(phito(xeqy)))
3, 7brpi21<8(lnotexistsxphitoexistsyforallx(phito(xeqy)))
1∃1→∃mo-09(_e1xphitoexistsyforallx(phito(xeqy)))
8, 9ja-pre10((existsxphito_e1xphi)toexistsyforallx(phito(xeqy)))
1df-∃1-alt311((existsxphiwedgeexistsyforallx(phito(xeqy)))to_e1xphi)
11exp12(existsxphito(existsyforallx(phito(xeqy))to_e1xphi))
12com12i13(existsyforallx(phito(xeqy))to(existsxphito_e1xphi))
10, 13>bii14((existsxphito_e1xphi)leftrightarrowexistsyforallx(phito(xeqy)))
2, 14bitr15(_em1xphileftrightarrowexistsyforallx(phito(xeqy)))