The usual: show that the argument is valid. The easiest way: assume a negation of the conclusion (indirect proof).
- (∃x)[Px • (y)(Py ⊃ y = x) • Qx]
- ¬ Qa
- ∴¬ Pa
- * ¬ ¬ Pa ......... AIP
- * Pa ......... 4 DN
- * Pm • (y)(Py ⊃ y = m) • Qm ......... 1 EI x/m
- * (y)(Py ⊃ y = m) ......... 6 Simp.
- * Pa ⊃ a = m ......... 7 UI y/a
- * a = m ......... 5,8 MP
- * ¬ Qm ......... 2, 9 Id
- * Qm ......... 6 Simp.
- * ¬ Qm • Qm ......... 10,11 Conj.
- ¬ ¬ ¬ Pa ......... 4-12 IP
- ¬ Pa ......... 13 DN
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