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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 4541 | . . . 4 | |
2 | th3q.1 | . . . . 5 | |
3 | 2 | ecelqsi 6451 | . . . 4 |
4 | 1, 3 | syl 14 | . . 3 |
5 | opelxpi 4541 | . . . 4 | |
6 | 2 | ecelqsi 6451 | . . . 4 |
7 | 5, 6 | syl 14 | . . 3 |
8 | 4, 7 | anim12i 336 | . 2 |
9 | eqid 2117 | . . . 4 | |
10 | eqid 2117 | . . . 4 | |
11 | 9, 10 | pm3.2i 270 | . . 3 |
12 | eqid 2117 | . . 3 | |
13 | opeq12 3677 | . . . . . 6 | |
14 | eceq1 6432 | . . . . . . . . 9 | |
15 | 14 | eqeq2d 2129 | . . . . . . . 8 |
16 | 15 | anbi1d 460 | . . . . . . 7 |
17 | oveq1 5749 | . . . . . . . . 9 | |
18 | 17 | eceq1d 6433 | . . . . . . . 8 |
19 | 18 | eqeq2d 2129 | . . . . . . 7 |
20 | 16, 19 | anbi12d 464 | . . . . . 6 |
21 | 13, 20 | syl 14 | . . . . 5 |
22 | 21 | spc2egv 2749 | . . . 4 |
23 | opeq12 3677 | . . . . . . 7 | |
24 | eceq1 6432 | . . . . . . . . . 10 | |
25 | 24 | eqeq2d 2129 | . . . . . . . . 9 |
26 | 25 | anbi2d 459 | . . . . . . . 8 |
27 | oveq2 5750 | . . . . . . . . . 10 | |
28 | 27 | eceq1d 6433 | . . . . . . . . 9 |
29 | 28 | eqeq2d 2129 | . . . . . . . 8 |
30 | 26, 29 | anbi12d 464 | . . . . . . 7 |
31 | 23, 30 | syl 14 | . . . . . 6 |
32 | 31 | spc2egv 2749 | . . . . 5 |
33 | 32 | 2eximdv 1838 | . . . 4 |
34 | 22, 33 | sylan9 406 | . . 3 |
35 | 11, 12, 34 | mp2ani 428 | . 2 |
36 | ecexg 6401 | . . . 4 | |
37 | 2, 36 | ax-mp 5 | . . 3 |
38 | eqeq1 2124 | . . . . . . . 8 | |
39 | eqeq1 2124 | . . . . . . . 8 | |
40 | 38, 39 | bi2anan9 580 | . . . . . . 7 |
41 | eqeq1 2124 | . . . . . . 7 | |
42 | 40, 41 | bi2anan9 580 | . . . . . 6 |
43 | 42 | 3impa 1161 | . . . . 5 |
44 | 43 | 4exbidv 1826 | . . . 4 |
45 | th3q.2 | . . . . 5 | |
46 | th3q.4 | . . . . 5 | |
47 | 2, 45, 46 | th3qlem2 6500 | . . . 4 |
48 | th3q.5 | . . . 4 | |
49 | 44, 47, 48 | ovig 5860 | . . 3 |
50 | 37, 49 | mp3an3 1289 | . 2 |
51 | 8, 35, 50 | sylc 62 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 947 wceq 1316 wex 1453 wcel 1465 cvv 2660 cop 3500 class class class wbr 3899 cxp 4507 (class class class)co 5742 coprab 5743 wer 6394 cec 6395 cqs 6396 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fv 5101 df-ov 5745 df-oprab 5746 df-er 6397 df-ec 6399 df-qs 6403 |
This theorem is referenced by: oviec 6503 |
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