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Mirrors > Home > ILE Home > Th. List > th3qlem2 | Unicode version |
Description: Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60, extended to operations on ordered pairs. The fourth hypothesis is the compatibility assumption. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
th3q.1 | |
th3q.2 | |
th3q.4 |
Ref | Expression |
---|---|
th3qlem2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | th3q.2 | . . 3 | |
2 | eqid 2117 | . . . . 5 | |
3 | breq1 3902 | . . . . . . . 8 | |
4 | 3 | anbi1d 460 | . . . . . . 7 |
5 | oveq1 5749 | . . . . . . . 8 | |
6 | 5 | breq1d 3909 | . . . . . . 7 |
7 | 4, 6 | imbi12d 233 | . . . . . 6 |
8 | 7 | imbi2d 229 | . . . . 5 |
9 | breq2 3903 | . . . . . . . 8 | |
10 | 9 | anbi1d 460 | . . . . . . 7 |
11 | oveq1 5749 | . . . . . . . 8 | |
12 | 11 | breq2d 3911 | . . . . . . 7 |
13 | 10, 12 | imbi12d 233 | . . . . . 6 |
14 | 13 | imbi2d 229 | . . . . 5 |
15 | breq1 3902 | . . . . . . . . . 10 | |
16 | 15 | anbi2d 459 | . . . . . . . . 9 |
17 | oveq2 5750 | . . . . . . . . . 10 | |
18 | 17 | breq1d 3909 | . . . . . . . . 9 |
19 | 16, 18 | imbi12d 233 | . . . . . . . 8 |
20 | 19 | imbi2d 229 | . . . . . . 7 |
21 | breq2 3903 | . . . . . . . . . 10 | |
22 | 21 | anbi2d 459 | . . . . . . . . 9 |
23 | oveq2 5750 | . . . . . . . . . 10 | |
24 | 23 | breq2d 3911 | . . . . . . . . 9 |
25 | 22, 24 | imbi12d 233 | . . . . . . . 8 |
26 | 25 | imbi2d 229 | . . . . . . 7 |
27 | th3q.4 | . . . . . . . 8 | |
28 | 27 | expcom 115 | . . . . . . 7 |
29 | 2, 20, 26, 28 | 2optocl 4586 | . . . . . 6 |
30 | 29 | com12 30 | . . . . 5 |
31 | 2, 8, 14, 30 | 2optocl 4586 | . . . 4 |
32 | 31 | imp 123 | . . 3 |
33 | 1, 32 | th3qlem1 6499 | . 2 |
34 | vex 2663 | . . . . . . 7 | |
35 | vex 2663 | . . . . . . 7 | |
36 | 34, 35 | opex 4121 | . . . . . 6 |
37 | vex 2663 | . . . . . . 7 | |
38 | vex 2663 | . . . . . . 7 | |
39 | 37, 38 | opex 4121 | . . . . . 6 |
40 | eceq1 6432 | . . . . . . . . 9 | |
41 | 40 | eqeq2d 2129 | . . . . . . . 8 |
42 | eceq1 6432 | . . . . . . . . 9 | |
43 | 42 | eqeq2d 2129 | . . . . . . . 8 |
44 | 41, 43 | bi2anan9 580 | . . . . . . 7 |
45 | oveq12 5751 | . . . . . . . . 9 | |
46 | 45 | eceq1d 6433 | . . . . . . . 8 |
47 | 46 | eqeq2d 2129 | . . . . . . 7 |
48 | 44, 47 | anbi12d 464 | . . . . . 6 |
49 | 36, 39, 48 | spc2ev 2755 | . . . . 5 |
50 | 49 | exlimivv 1852 | . . . 4 |
51 | 50 | exlimivv 1852 | . . 3 |
52 | 51 | moimi 2042 | . 2 |
53 | 33, 52 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wex 1453 wcel 1465 wmo 1978 cvv 2660 cop 3500 class class class wbr 3899 cxp 4507 (class class class)co 5742 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-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 |
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-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-fv 5101 df-ov 5745 df-er 6397 df-ec 6399 df-qs 6403 |
This theorem is referenced by: th3qcor 6501 th3q 6502 |
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