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Does anyone know how to solve 13.49

∃x P(x)
∀x ∀y ((P(x) ∧ P(y)) → x = y)

= ∃x (P(x) ∧ ∀y (P(y) → y = x))

and 13.50

∃x (P(x) ∧ ∀y (P(y) → y = x))

= ∀x ∀y ((P(x) ∧ P(y)) → x = y)

I have big problems!

closed as off-topic by Mauro ALLEGRANZA, Keelan Apr 20 '17 at 7:47

This question appears to be off-topic. The users who voted to close gave this specific reason:

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  • 1
    First problem. Ask one question per question... Second problem, the formatting (I can fix the second problem) – virmaior Apr 19 '17 at 11:15
  • Quite similar to this post – Mauro ALLEGRANZA Apr 19 '17 at 12:16
  • Also, when referring to exercise numbers, you should mention the book you're taking them from. – Keelan Apr 19 '17 at 19:45
1
{1}      1.  Ǝx[Px]                            Prem.
{2}      2.  ∀x∀y[(Px & Py) → x=y]            Prem.
{3}      3.  Pa                                Assum.
{2}      4.  ∀y[(Pa & Py) → a=y]               2 UE
{5}      5.  Pb                                Assum. TD(b)
{3,5}    6.  Pa & Pb                           3,5 &I
{2}      7.  (Pa & Pb) → a=b                   4 UE
{2,3,5}  8.  a=b                               6,7 MP
{2,5}    9.  Pa → a=b                          3,8 CI
{2,5}    10. ∀y[Py → y=b]                     9 UI
{2,5}    11. Pb & ∀y[Py → y=b]                5,10 &I
{2,5}    12. Ǝx[Px & ∀y[Py → y=x]]            11 EI
{1,2}    13. Ǝx[Px & ∀y[Py → y=x]]            1,5,12 EE

Here's the second one:

{1}      1.  Ǝx[Px & ∀y[Py → y=x]]            Prem.
{2}      2.  Pa & Pb                          Assum.
{3}      3.  Pc & ∀y[Py → y=c]                Assum. TD(c)
{3}      4.  ∀y[Py → y=c]                     3 &E
{3}      5.  Pa → a=c                         4 UE
{2}      6.  Pa                               2 &E
{2,3}    7.  a=c                              5,6 MP
{3}      8.  Pb → b=c                         4 UE
{2}      9.  Pb                               2 &E
{2,3}    10. b=c                              8,9 MP
{2,3}    11. a=b                              7,10 =E
{3}      12. (Pa & Pb) → a=b                  2,11 MP
{3}      13. ∀y[(Pa & Py) → a=y]              12 UI
{3}      14. ∀x∀y[(Px & Py) → x=y]            13 UI
{1}      i5. ∀x∀y[(Px & Py) → x=y]            1,3,14 EE

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