Destructive Dilemma/Formulation 2

Theorem

$\ds \paren {p \implies q} \land \paren {r \implies s}$

$\ds $

$\ds \neg q \lor \neg s$

$\ds $

$\ds \vdash \ \ $

$\ds \neg p \lor \neg r$

$\ds $

$\paren {p \implies q} \land \paren {r \implies s}, \neg q \lor \neg s \vdash \neg p \lor \neg r$
Line	Pool	Formula	Rule	Depends upon	Notes
1	1	$\paren {p \implies q} \land \paren {r \implies s}$	Premise	(None)
2	2	$\neg q \lor \neg s$	Premise	(None)
3	1	$p \implies q$	Rule of Simplification: $\land \EE_1$	1
4	1	$r \implies s$	Rule of Simplification: $\land \EE_2$	1
5	5	$\neg q$	Assumption	(None)
6	1, 5	$\neg p$	Modus Tollendo Tollens (MTT)	3, 5
7	1, 5	$\neg p \lor \neg r$	Rule of Addition: $\lor \II_1$	6
8	8	$\neg s$	Assumption	(None)
9	1, 8	$\neg r$	Modus Tollendo Tollens (MTT)	4, 8
10	1, 8	$\neg p \lor \neg r$	Rule of Addition: $\lor \II_2$	9
11	1, 2	$\neg p \lor \neg r$	Proof by Cases: $\text{PBC}$	2, 5 – 7, 8 – 10	Assumptions 5 and 8 have been discharged

$\blacksquare$

1973: Irving M. Copi: Symbolic Logic (4th ed.) ... (previous) ... (next): $2$ Arguments Containing Compound Statements: $2.3$: Argument Forms and Truth Tables: Exercise $\text{II} \ \mathbf {18}.$
1973: Irving M. Copi: Symbolic Logic (4th ed.) ... (previous) ... (next): $3$: The Method of Deduction: $3.1$: Formal Proof of Validity: Rules of Inference: $6.$
1980: D.J. O'Connor and Betty Powell: Elementary Logic ... (previous) ... (next): $\S \text{II}$: The Logic of Statements $(2): \ 1$: Decision procedures and proofs: $4$