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Mirrors > Home > ILE Home > Th. List > intab | Unicode version |
Description: The intersection of a special case of a class abstraction. may be free in and , which can be thought of a and . (Contributed by NM, 28-Jul-2006.) (Proof shortened by Mario Carneiro, 14-Nov-2016.) |
Ref | Expression |
---|---|
intab.1 | |
intab.2 |
Ref | Expression |
---|---|
intab |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2177 | . . . . . . . . . 10 | |
2 | 1 | anbi2d 461 | . . . . . . . . 9 |
3 | 2 | exbidv 1818 | . . . . . . . 8 |
4 | 3 | cbvabv 2295 | . . . . . . 7 |
5 | intab.2 | . . . . . . 7 | |
6 | 4, 5 | eqeltri 2243 | . . . . . 6 |
7 | nfe1 1489 | . . . . . . . . 9 | |
8 | 7 | nfab 2317 | . . . . . . . 8 |
9 | 8 | nfeq2 2324 | . . . . . . 7 |
10 | eleq2 2234 | . . . . . . . 8 | |
11 | 10 | imbi2d 229 | . . . . . . 7 |
12 | 9, 11 | albid 1608 | . . . . . 6 |
13 | 6, 12 | elab 2874 | . . . . 5 |
14 | 19.8a 1583 | . . . . . . . . 9 | |
15 | 14 | ex 114 | . . . . . . . 8 |
16 | 15 | alrimiv 1867 | . . . . . . 7 |
17 | intab.1 | . . . . . . . 8 | |
18 | 17 | sbc6 2980 | . . . . . . 7 |
19 | 16, 18 | sylibr 133 | . . . . . 6 |
20 | df-sbc 2956 | . . . . . 6 | |
21 | 19, 20 | sylib 121 | . . . . 5 |
22 | 13, 21 | mpgbir 1446 | . . . 4 |
23 | intss1 3844 | . . . 4 | |
24 | 22, 23 | ax-mp 5 | . . 3 |
25 | 19.29r 1614 | . . . . . . . 8 | |
26 | simplr 525 | . . . . . . . . . 10 | |
27 | pm3.35 345 | . . . . . . . . . . 11 | |
28 | 27 | adantlr 474 | . . . . . . . . . 10 |
29 | 26, 28 | eqeltrd 2247 | . . . . . . . . 9 |
30 | 29 | exlimiv 1591 | . . . . . . . 8 |
31 | 25, 30 | syl 14 | . . . . . . 7 |
32 | 31 | ex 114 | . . . . . 6 |
33 | 32 | alrimiv 1867 | . . . . 5 |
34 | vex 2733 | . . . . . 6 | |
35 | 34 | elintab 3840 | . . . . 5 |
36 | 33, 35 | sylibr 133 | . . . 4 |
37 | 36 | abssi 3222 | . . 3 |
38 | 24, 37 | eqssi 3163 | . 2 |
39 | 38, 4 | eqtri 2191 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wal 1346 wceq 1348 wex 1485 wcel 2141 cab 2156 cvv 2730 wsbc 2955 wss 3121 cint 3829 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-sbc 2956 df-in 3127 df-ss 3134 df-int 3830 |
This theorem is referenced by: (None) |
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