Change the structure from 3SAT to 1in3 3SAT
A 1-in-3 SAT clause (A,B,C) can be expressed with the following 5 clauses in 3SAT. 

$(A ∨ B ∨ C) $

$(A ∨ \overline{B} ∨ \overline{C}) $

$(\overline{A} ∨ B ∨ \overline{C}) $

$(\overline{A} ∨ \overline{B} ∨ C) $

$(\overline{A} ∨ \overline{B} ∨ \overline{C}) $

Why? Consider the truth table over A,B,C. 

| A | B | C |   | A | B | C |
| 0 | 0 | 0 |   | --------- |
| 0 | 0 | 1 |   | 0 | 0 | 1 | 
| 0 | 1 | 0 |   | 0 | 1 | 0 | 
| 0 | 1 | 1 |   | --------- | 
| 1 | 0 | 0 |   | 1 | 0 | 0 |
| 1 | 0 | 1 |   | --------- | 
| 1 | 1 | 0 |   | --------- |
| 1 | 1 | 1 |   | --------- |


There are 8 lines. Each clause eliminates 1 line. We only want to allow (0,0,1), (0,1,0), (1,0,0) and eliminate the others.