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Mirrors > Home > ILE Home > Th. List > th3q | Unicode version |
Description: Theorem 3Q of [Enderton] p. 60, extended to operations on ordered pairs. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 19-Dec-2013.) |
Ref | Expression |
---|---|
th3q.1 | |
th3q.2 | |
th3q.4 | |
th3q.5 |
Ref | Expression |
---|---|
th3q |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxpi 4376 | . . . 4 | |
2 | th3q.1 | . . . . 5 | |
3 | 2 | ecelqsi 6160 | . . . 4 |
4 | 1, 3 | syl 14 | . . 3 |
5 | opelxpi 4376 | . . . 4 | |
6 | 2 | ecelqsi 6160 | . . . 4 |
7 | 5, 6 | syl 14 | . . 3 |
8 | 4, 7 | anim12i 321 | . 2 |
9 | eqid 2040 | . . . 4 | |
10 | eqid 2040 | . . . 4 | |
11 | 9, 10 | pm3.2i 257 | . . 3 |
12 | eqid 2040 | . . 3 | |
13 | opeq12 3551 | . . . . . 6 | |
14 | eceq1 6141 | . . . . . . . . 9 | |
15 | 14 | eqeq2d 2051 | . . . . . . . 8 |
16 | 15 | anbi1d 438 | . . . . . . 7 |
17 | oveq1 5519 | . . . . . . . . 9 | |
18 | 17 | eceq1d 6142 | . . . . . . . 8 |
19 | 18 | eqeq2d 2051 | . . . . . . 7 |
20 | 16, 19 | anbi12d 442 | . . . . . 6 |
21 | 13, 20 | syl 14 | . . . . 5 |
22 | 21 | spc2egv 2642 | . . . 4 |
23 | opeq12 3551 | . . . . . . 7 | |
24 | eceq1 6141 | . . . . . . . . . 10 | |
25 | 24 | eqeq2d 2051 | . . . . . . . . 9 |
26 | 25 | anbi2d 437 | . . . . . . . 8 |
27 | oveq2 5520 | . . . . . . . . . 10 | |
28 | 27 | eceq1d 6142 | . . . . . . . . 9 |
29 | 28 | eqeq2d 2051 | . . . . . . . 8 |
30 | 26, 29 | anbi12d 442 | . . . . . . 7 |
31 | 23, 30 | syl 14 | . . . . . 6 |
32 | 31 | spc2egv 2642 | . . . . 5 |
33 | 32 | 2eximdv 1762 | . . . 4 |
34 | 22, 33 | sylan9 389 | . . 3 |
35 | 11, 12, 34 | mp2ani 408 | . 2 |
36 | ecexg 6110 | . . . 4 | |
37 | 2, 36 | ax-mp 7 | . . 3 |
38 | eqeq1 2046 | . . . . . . . 8 | |
39 | eqeq1 2046 | . . . . . . . 8 | |
40 | 38, 39 | bi2anan9 538 | . . . . . . 7 |
41 | eqeq1 2046 | . . . . . . 7 | |
42 | 40, 41 | bi2anan9 538 | . . . . . 6 |
43 | 42 | 3impa 1099 | . . . . 5 |
44 | 43 | 4exbidv 1750 | . . . 4 |
45 | th3q.2 | . . . . 5 | |
46 | th3q.4 | . . . . 5 | |
47 | 2, 45, 46 | th3qlem2 6209 | . . . 4 |
48 | th3q.5 | . . . 4 | |
49 | 44, 47, 48 | ovig 5622 | . . 3 |
50 | 37, 49 | mp3an3 1221 | . 2 |
51 | 8, 35, 50 | sylc 56 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 w3a 885 wceq 1243 wex 1381 wcel 1393 cvv 2557 cop 3378 class class class wbr 3764 cxp 4343 (class class class)co 5512 coprab 5513 wer 6103 cec 6104 cqs 6105 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fv 4910 df-ov 5515 df-oprab 5516 df-er 6106 df-ec 6108 df-qs 6112 |
This theorem is referenced by: oviec 6212 |
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