Description: There is a line through any two distinct points. Hilbert's axiom I.1 for geometry. (Contributed by Scott Fenton, 29-Oct-2013) (Revised by Mario Carneiro, 19-Apr-2014)
Ref | Expression | ||
---|---|---|---|
Assertion | hilbert1.1 | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 | |
|
2 | simp2 | |
|
3 | simp3 | |
|
4 | eqidd | |
|
5 | neeq1 | |
|
6 | oveq1 | |
|
7 | 6 | eqeq2d | |
8 | 5 7 | anbi12d | |
9 | neeq2 | |
|
10 | oveq2 | |
|
11 | 10 | eqeq2d | |
12 | 9 11 | anbi12d | |
13 | 8 12 | rspc2ev | |
14 | 1 2 3 4 13 | syl112anc | |
15 | fveq2 | |
|
16 | 15 | rexeqdv | |
17 | 15 16 | rexeqbidv | |
18 | 17 | rspcev | |
19 | 14 18 | sylan2 | |
20 | ellines | |
|
21 | 19 20 | sylibr | |
22 | linerflx1 | |
|
23 | linerflx2 | |
|
24 | eleq2 | |
|
25 | eleq2 | |
|
26 | 24 25 | anbi12d | |
27 | 26 | rspcev | |
28 | 21 22 23 27 | syl12anc | |