Description: Specify the same property in two ways when class B ( y ) is single-valued. (Contributed by NM, 1-Nov-2010) (Proof shortened by Mario Carneiro, 24-Dec-2016)
Ref | Expression | ||
---|---|---|---|
Hypothesis | eusvobj1.1 | |
|
Assertion | eusvobj2 | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eusvobj1.1 | |
|
2 | euabsn2 | |
|
3 | eleq2 | |
|
4 | abid | |
|
5 | velsn | |
|
6 | 3 4 5 | 3bitr3g | |
7 | nfre1 | |
|
8 | 7 | nfab | |
9 | 8 | nfeq1 | |
10 | 1 | elabrex | |
11 | eleq2 | |
|
12 | 1 | elsn | |
13 | eqcom | |
|
14 | 12 13 | bitri | |
15 | 11 14 | bitrdi | |
16 | 10 15 | syl5ib | |
17 | 9 16 | ralrimi | |
18 | eqeq1 | |
|
19 | 18 | ralbidv | |
20 | 17 19 | syl5ibrcom | |
21 | 6 20 | sylbid | |
22 | 21 | exlimiv | |
23 | 2 22 | sylbi | |
24 | euex | |
|
25 | rexn0 | |
|
26 | 25 | exlimiv | |
27 | r19.2z | |
|
28 | 27 | ex | |
29 | 24 26 28 | 3syl | |
30 | 23 29 | impbid | |