"If the teller or the cashier had pushed the alarm button, the vault would have locked automatically, and the police would have arrived within three minutes. Had the police arrived in three minutes, the robbers’ car would have been overtaken. However, the robbers’ car was not overtaken. Therefore the teller did not push the alarm button. [ Let T"the teller pushed the alarm button", C"the cashier pushed the alarm button", V"the vault locked automatically", P"the police arrived within three minutes", O"the robbers’ car was overtaken" ]

Complete Question:

Translate this valid argument into symbolic form, then prove it using rules of inference/identities. "If the teller or the cashier had pushed the alarm button, the vault would have locked automatically, and the police would have arrived within three minutes. Had the police arrived in three minutes, the robbers' car would have been overtaken. However, the robbers' car was not overtaken. Therefore the teller did not push the alarm button. [ Let T="the teller pushed the alarm button", C="the cashier pushed the alarm button", V="the vault locked automatically", P="the police arrived within three minutes", 0="the robbers' car was overtaken" ]

1) (T v C) => (V Λ P )

2) P => O

3) ⌐ O                          / ∴ ⌐ T

4) ⌐ P                           2, 3, M.T.

5 ⌐ P v ⌐V                    4, Add.

6 ⌐V v ⌐ P                     5,Com.

7 ⌐ (T v C )                     1,6. M.T

8 ⌐ T Λ ⌐ C                   7, De.M.

9∴ ⌐T                                8, Simp.