dfsbf

Theorem.

Arguments:

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

Dummy variables:

w (sv),

Distinct variable conditions:

(w, y), (w, x), (w, phi), (w, z), (z, y), (z, x),

Hypotheses:

(z bound in phi)

Assertions:

([y/x]phileftrightarrowexistsz((zeqy)wedgeexistsx((xeqz)wedgephi)))

Proof:

Hyp Ref Line Expr
Hypo1(z bound in phi)
df-sb2([y/x]phileftrightarrowexistsw((weqy)wedgeexistsx((xeqw)wedgephi)))
eqt>3((weqz)to((xeqw)leftrightarrow(xeqz)))
3bi.bldan<d4((weqz)to(((xeqw)wedgephi)leftrightarrow((xeqz)wedgephi)))
ax17-bv5(x bound in (weqz))
4, 5bi.bldexd6((weqz)to(existsx((xeqw)wedgephi)leftrightarrowexistsx((xeqz)wedgephi)))
eqt<7((weqz)to((weqy)leftrightarrow(zeqy)))
6, 7bi.bldan<>d8((weqz)to(((weqy)wedgeexistsx((xeqw)wedgephi))leftrightarrow((zeqy)wedgeexistsx((xeqz)wedgephi))))
ax17-bv9(w bound in ((zeqy)wedgeexistsx((xeqz)wedgephi)))
ax17-bv10(z bound in (weqy))
ax17-bv11(z bound in (xeqw))
1, 11bv-∧12(z bound in ((xeqw)wedgephi))
12bv-∃13(z bound in existsx((xeqw)wedgephi))
10, 13bv-∧14(z bound in ((weqy)wedgeexistsx((xeqw)wedgephi)))
8, 9, 14cbvex15(existsw((weqy)wedgeexistsx((xeqw)wedgephi))leftrightarrowexistsz((zeqy)wedgeexistsx((xeqz)wedgephi)))
2, 15bitr16([y/x]phileftrightarrowexistsz((zeqy)wedgeexistsx((xeqz)wedgephi)))