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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 4571 | . . . 4 | |
2 | th3q.1 | . . . . 5 | |
3 | 2 | ecelqsi 6483 | . . . 4 |
4 | 1, 3 | syl 14 | . . 3 |
5 | opelxpi 4571 | . . . 4 | |
6 | 2 | ecelqsi 6483 | . . . 4 |
7 | 5, 6 | syl 14 | . . 3 |
8 | 4, 7 | anim12i 336 | . 2 |
9 | eqid 2139 | . . . 4 | |
10 | eqid 2139 | . . . 4 | |
11 | 9, 10 | pm3.2i 270 | . . 3 |
12 | eqid 2139 | . . 3 | |
13 | opeq12 3707 | . . . . . 6 | |
14 | eceq1 6464 | . . . . . . . . 9 | |
15 | 14 | eqeq2d 2151 | . . . . . . . 8 |
16 | 15 | anbi1d 460 | . . . . . . 7 |
17 | oveq1 5781 | . . . . . . . . 9 | |
18 | 17 | eceq1d 6465 | . . . . . . . 8 |
19 | 18 | eqeq2d 2151 | . . . . . . 7 |
20 | 16, 19 | anbi12d 464 | . . . . . 6 |
21 | 13, 20 | syl 14 | . . . . 5 |
22 | 21 | spc2egv 2775 | . . . 4 |
23 | opeq12 3707 | . . . . . . 7 | |
24 | eceq1 6464 | . . . . . . . . . 10 | |
25 | 24 | eqeq2d 2151 | . . . . . . . . 9 |
26 | 25 | anbi2d 459 | . . . . . . . 8 |
27 | oveq2 5782 | . . . . . . . . . 10 | |
28 | 27 | eceq1d 6465 | . . . . . . . . 9 |
29 | 28 | eqeq2d 2151 | . . . . . . . 8 |
30 | 26, 29 | anbi12d 464 | . . . . . . 7 |
31 | 23, 30 | syl 14 | . . . . . 6 |
32 | 31 | spc2egv 2775 | . . . . 5 |
33 | 32 | 2eximdv 1854 | . . . 4 |
34 | 22, 33 | sylan9 406 | . . 3 |
35 | 11, 12, 34 | mp2ani 428 | . 2 |
36 | ecexg 6433 | . . . 4 | |
37 | 2, 36 | ax-mp 5 | . . 3 |
38 | eqeq1 2146 | . . . . . . . 8 | |
39 | eqeq1 2146 | . . . . . . . 8 | |
40 | 38, 39 | bi2anan9 595 | . . . . . . 7 |
41 | eqeq1 2146 | . . . . . . 7 | |
42 | 40, 41 | bi2anan9 595 | . . . . . 6 |
43 | 42 | 3impa 1176 | . . . . 5 |
44 | 43 | 4exbidv 1842 | . . . 4 |
45 | th3q.2 | . . . . 5 | |
46 | th3q.4 | . . . . 5 | |
47 | 2, 45, 46 | th3qlem2 6532 | . . . 4 |
48 | th3q.5 | . . . 4 | |
49 | 44, 47, 48 | ovig 5892 | . . 3 |
50 | 37, 49 | mp3an3 1304 | . 2 |
51 | 8, 35, 50 | sylc 62 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wex 1468 wcel 1480 cvv 2686 cop 3530 class class class wbr 3929 cxp 4537 (class class class)co 5774 coprab 5775 wer 6426 cec 6427 cqs 6428 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fv 5131 df-ov 5777 df-oprab 5778 df-er 6429 df-ec 6431 df-qs 6435 |
This theorem is referenced by: oviec 6535 |
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