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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 4630 | . . . 4 | |
2 | th3q.1 | . . . . 5 | |
3 | 2 | ecelqsi 6546 | . . . 4 |
4 | 1, 3 | syl 14 | . . 3 |
5 | opelxpi 4630 | . . . 4 | |
6 | 2 | ecelqsi 6546 | . . . 4 |
7 | 5, 6 | syl 14 | . . 3 |
8 | 4, 7 | anim12i 336 | . 2 |
9 | eqid 2164 | . . . 4 | |
10 | eqid 2164 | . . . 4 | |
11 | 9, 10 | pm3.2i 270 | . . 3 |
12 | eqid 2164 | . . 3 | |
13 | opeq12 3754 | . . . . . 6 | |
14 | eceq1 6527 | . . . . . . . . 9 | |
15 | 14 | eqeq2d 2176 | . . . . . . . 8 |
16 | 15 | anbi1d 461 | . . . . . . 7 |
17 | oveq1 5843 | . . . . . . . . 9 | |
18 | 17 | eceq1d 6528 | . . . . . . . 8 |
19 | 18 | eqeq2d 2176 | . . . . . . 7 |
20 | 16, 19 | anbi12d 465 | . . . . . 6 |
21 | 13, 20 | syl 14 | . . . . 5 |
22 | 21 | spc2egv 2811 | . . . 4 |
23 | opeq12 3754 | . . . . . . 7 | |
24 | eceq1 6527 | . . . . . . . . . 10 | |
25 | 24 | eqeq2d 2176 | . . . . . . . . 9 |
26 | 25 | anbi2d 460 | . . . . . . . 8 |
27 | oveq2 5844 | . . . . . . . . . 10 | |
28 | 27 | eceq1d 6528 | . . . . . . . . 9 |
29 | 28 | eqeq2d 2176 | . . . . . . . 8 |
30 | 26, 29 | anbi12d 465 | . . . . . . 7 |
31 | 23, 30 | syl 14 | . . . . . 6 |
32 | 31 | spc2egv 2811 | . . . . 5 |
33 | 32 | 2eximdv 1869 | . . . 4 |
34 | 22, 33 | sylan9 407 | . . 3 |
35 | 11, 12, 34 | mp2ani 429 | . 2 |
36 | ecexg 6496 | . . . 4 | |
37 | 2, 36 | ax-mp 5 | . . 3 |
38 | eqeq1 2171 | . . . . . . . 8 | |
39 | eqeq1 2171 | . . . . . . . 8 | |
40 | 38, 39 | bi2anan9 596 | . . . . . . 7 |
41 | eqeq1 2171 | . . . . . . 7 | |
42 | 40, 41 | bi2anan9 596 | . . . . . 6 |
43 | 42 | 3impa 1183 | . . . . 5 |
44 | 43 | 4exbidv 1857 | . . . 4 |
45 | th3q.2 | . . . . 5 | |
46 | th3q.4 | . . . . 5 | |
47 | 2, 45, 46 | th3qlem2 6595 | . . . 4 |
48 | th3q.5 | . . . 4 | |
49 | 44, 47, 48 | ovig 5954 | . . 3 |
50 | 37, 49 | mp3an3 1315 | . 2 |
51 | 8, 35, 50 | sylc 62 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 967 wceq 1342 wex 1479 wcel 2135 cvv 2721 cop 3573 class class class wbr 3976 cxp 4596 (class class class)co 5836 coprab 5837 wer 6489 cec 6490 cqs 6491 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fv 5190 df-ov 5839 df-oprab 5840 df-er 6492 df-ec 6494 df-qs 6498 |
This theorem is referenced by: oviec 6598 |
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