eqt-sb

Theorem.

Arguments:

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

Dummy variables:

NIL (sv),

Distinct variable conditions:

(NIL, x), (NIL, y), (NIL, z), (NIL, phi),

Assertions:

((xeqy)to([x/z]phileftrightarrow[y/z]phi))

Proof:

Hyp Ref Line Expr
eqt>1((xeqy)to((NILeqx)leftrightarrow(NILeqy)))
ax17-bv2(NIL bound in (xeqy))
1bi.bldan<d3((xeqy)to(((NILeqx)wedgeexistsz((zeqNIL)wedgephi))leftrightarrow((NILeqy)wedgeexistsz((zeqNIL)wedgephi))))
2, 3bi.bldexd4((xeqy)to(existsNIL((NILeqx)wedgeexistsz((zeqNIL)wedgephi))leftrightarrowexistsNIL((NILeqy)wedgeexistsz((zeqNIL)wedgephi))))
df-sb5([x/z]phileftrightarrowexistsNIL((NILeqx)wedgeexistsz((zeqNIL)wedgephi)))
df-sb6([y/z]phileftrightarrowexistsNIL((NILeqy)wedgeexistsz((zeqNIL)wedgephi)))
4, 5, 6<2bitrg7((xeqy)to([x/z]phileftrightarrow[y/z]phi))