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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 2040 | . . . . 5 | |
3 | breq1 3767 | . . . . . . . 8 | |
4 | 3 | anbi1d 438 | . . . . . . 7 |
5 | oveq1 5519 | . . . . . . . 8 | |
6 | 5 | breq1d 3774 | . . . . . . 7 |
7 | 4, 6 | imbi12d 223 | . . . . . 6 |
8 | 7 | imbi2d 219 | . . . . 5 |
9 | breq2 3768 | . . . . . . . 8 | |
10 | 9 | anbi1d 438 | . . . . . . 7 |
11 | oveq1 5519 | . . . . . . . 8 | |
12 | 11 | breq2d 3776 | . . . . . . 7 |
13 | 10, 12 | imbi12d 223 | . . . . . 6 |
14 | 13 | imbi2d 219 | . . . . 5 |
15 | breq1 3767 | . . . . . . . . . 10 | |
16 | 15 | anbi2d 437 | . . . . . . . . 9 |
17 | oveq2 5520 | . . . . . . . . . 10 | |
18 | 17 | breq1d 3774 | . . . . . . . . 9 |
19 | 16, 18 | imbi12d 223 | . . . . . . . 8 |
20 | 19 | imbi2d 219 | . . . . . . 7 |
21 | breq2 3768 | . . . . . . . . . 10 | |
22 | 21 | anbi2d 437 | . . . . . . . . 9 |
23 | oveq2 5520 | . . . . . . . . . 10 | |
24 | 23 | breq2d 3776 | . . . . . . . . 9 |
25 | 22, 24 | imbi12d 223 | . . . . . . . 8 |
26 | 25 | imbi2d 219 | . . . . . . 7 |
27 | th3q.4 | . . . . . . . 8 | |
28 | 27 | expcom 109 | . . . . . . 7 |
29 | 2, 20, 26, 28 | 2optocl 4417 | . . . . . 6 |
30 | 29 | com12 27 | . . . . 5 |
31 | 2, 8, 14, 30 | 2optocl 4417 | . . . 4 |
32 | 31 | imp 115 | . . 3 |
33 | 1, 32 | th3qlem1 6208 | . 2 |
34 | vex 2560 | . . . . . . 7 | |
35 | vex 2560 | . . . . . . 7 | |
36 | 34, 35 | opex 3966 | . . . . . 6 |
37 | vex 2560 | . . . . . . 7 | |
38 | vex 2560 | . . . . . . 7 | |
39 | 37, 38 | opex 3966 | . . . . . 6 |
40 | eceq1 6141 | . . . . . . . . 9 | |
41 | 40 | eqeq2d 2051 | . . . . . . . 8 |
42 | eceq1 6141 | . . . . . . . . 9 | |
43 | 42 | eqeq2d 2051 | . . . . . . . 8 |
44 | 41, 43 | bi2anan9 538 | . . . . . . 7 |
45 | oveq12 5521 | . . . . . . . . 9 | |
46 | 45 | eceq1d 6142 | . . . . . . . 8 |
47 | 46 | eqeq2d 2051 | . . . . . . 7 |
48 | 44, 47 | anbi12d 442 | . . . . . 6 |
49 | 36, 39, 48 | spc2ev 2648 | . . . . 5 |
50 | 49 | exlimivv 1776 | . . . 4 |
51 | 50 | exlimivv 1776 | . . 3 |
52 | 51 | moimi 1965 | . 2 |
53 | 33, 52 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 wex 1381 wcel 1393 wmo 1901 cvv 2557 cop 3378 class class class wbr 3764 cxp 4343 (class class class)co 5512 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-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 |
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-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-fv 4910 df-ov 5515 df-er 6106 df-ec 6108 df-qs 6112 |
This theorem is referenced by: th3qcor 6210 th3q 6211 |
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