Question 188825
Here's one way to do it...


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1. p <--> q
2. p -> r
3. p -> (r -> s) Therefore: q -> s
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4.  (p -> q) & (q -> p)                      1     Material Equivalence
5.  (q -> p) & (p -> q)                      4     Commutation
6.  q -> p                                   5     Simplification
7.  q -> r                                   6,2   Hypothetical Syllogism
8.  q -> (r -> s)                            6,3   Hypothetical Syllogism
9.  (q & r) -> s                             8     Exportation
10. ~(q & r) v s                             9     Material Implication    
11. (~q v ~r) v s                            10    DeMorgan's Theorem
12. ~q v (~r v s)                            11    Association
13. ~q v (s v ~r)                            12    Commutation
14. (~q v s) v ~r                            13    Association
15. (q -> s) v ~r                            14    Material Implication
16. ~r v (q -> s)                            15    Commutation
17. r -> (q -> s)                            16    Material Implication
18. q -> (q -> s)                            7,17  Hypothetical Syllogism
19. (q & q) -> s                             18    Exportation
20. q -> s                                   19    Tautology
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