chvar-∃

Theorem.

Arguments:

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

Hypotheses:

(y bound in phi)

Assertions:

(existsy[y/x]phileftrightarrowexistsxphi)

Proof:

Hyp Ref Line Expr
Hypo1(y bound in phi)
1bv-¬2(y bound in lnotphi)
sb-neg3([y/x]lnotphileftrightarrowlnot[y/x]phi)
3eqt-∀-i4(forally[y/x]lnotphileftrightarrowforallylnot[y/x]phi)
2chvar-∀5(forally[y/x]lnotphileftrightarrowforallxlnotphi)
4, 5<>bitr6(forallylnot[y/x]phileftrightarrowforallxlnotphi)
6bi.bldneg7(lnotforallylnot[y/x]phileftrightarrowlnotforallxlnotphi)
df-ex8(existsy[y/x]phileftrightarrowlnotforallylnot[y/x]phi)
df-ex9(existsxphileftrightarrowlnotforallxlnotphi)
7, 8, 9<2bitr10(existsy[y/x]phileftrightarrowexistsxphi)