$\forall x \forall y (T(x,y) \rightarrow x=y) \vdash T(a,b) \rightarrow T(b,a)$
1.
Playground
No Rule Found
1. $\forall x \forall y (T(x,y) \rightarrow x=y)$:PR
2. $\forall y (T(a,y) \rightarrow a=y)$: AE1
3. $\forall y (T(b,y) \rightarrow b=y)$: AE1
4. $T(a,b) \rightarrow a=b$: AE2
5. $T(a,b)$: AS
6. $\neg T(a,b)$: AS
7. $\bot$: $\neg$E 5, 6
8. $T(b,a)$: X7
9. $\neg T(a,b) \rightarrow T(b,a)$: $\rightarrow$I 6-8
10. $\neg T(a,b)$: AS
11. $T(b,a)$: $\rightarrow$E 9, 10
12. $T(a,b)$: R5
13. $\bot$: $\neg$E 10, 11
14.
15.
16.
17. $T(b,a)$:
18. $T(a,b) \rightarrow T(b,a) \rightarrow$I 5
1. $\forall x \forall y (T(x,y) \rightarrow x=y)$
PR
2. $\forall y (T(a,y) \rightarrow a=y)$
VE 1
3. $\forall y (T(b,y) \rightarrow b=y)$
VE 1
4. $T(a,b) \rightarrow a=b$
VE 2
5. $T(a,b)$
AS
6. $\neg T(a,b)$
AS
7. $\bot$
$\neg$E 5, 6
8. $T(b,a)$
X7
9. $\neg T(a,b) \rightarrow T(b,a)$
$\rightarrow$I 6-8
10. $\neg T(a,b)$
AS
11. $T(b,a)$
$\rightarrow$E 9, 10
12. $T(a,b)$
R5
13. $\bot$
$\neg$E 10, 11
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15.
16.
17. "T(b,a):"
18. "T(a,b)->T(b,a)->15"