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Mirrors > Home > ILE Home > Th. List > exmidontriimlem4 | Unicode version |
Description: Lemma for exmidontriim 7161. The induction step for the induction on . (Contributed by Jim Kingdon, 10-Aug-2024.) |
Ref | Expression |
---|---|
exmidontriimlem4.a | |
exmidontriimlem4.b | |
exmidontriimlem4.em | EXMID |
exmidontriimlem4.h |
Ref | Expression |
---|---|
exmidontriimlem4 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq2 2221 | . . 3 | |
2 | eqeq2 2167 | . . 3 | |
3 | eleq1 2220 | . . 3 | |
4 | 1, 2, 3 | 3orbi123d 1293 | . 2 |
5 | eleq2w 2219 | . . . . . . 7 | |
6 | eqeq2 2167 | . . . . . . 7 | |
7 | eleq1w 2218 | . . . . . . 7 | |
8 | 5, 6, 7 | 3orbi123d 1293 | . . . . . 6 |
9 | 8 | imbi2d 229 | . . . . 5 |
10 | exmidontriimlem4.a | . . . . . . . 8 | |
11 | 10 | adantl 275 | . . . . . . 7 |
12 | simpll 519 | . . . . . . 7 | |
13 | exmidontriimlem4.em | . . . . . . . 8 EXMID | |
14 | 13 | adantl 275 | . . . . . . 7 EXMID |
15 | exmidontriimlem4.h | . . . . . . . 8 | |
16 | 15 | adantl 275 | . . . . . . 7 |
17 | simplr 520 | . . . . . . . . . 10 | |
18 | eleq2w 2219 | . . . . . . . . . . . . 13 | |
19 | eqeq2 2167 | . . . . . . . . . . . . 13 | |
20 | eleq1w 2218 | . . . . . . . . . . . . 13 | |
21 | 18, 19, 20 | 3orbi123d 1293 | . . . . . . . . . . . 12 |
22 | 21 | imbi2d 229 | . . . . . . . . . . 11 |
23 | simpllr 524 | . . . . . . . . . . 11 | |
24 | simpr 109 | . . . . . . . . . . 11 | |
25 | 22, 23, 24 | rspcdva 2821 | . . . . . . . . . 10 |
26 | 17, 25 | mpd 13 | . . . . . . . . 9 |
27 | 26 | ralrimiva 2530 | . . . . . . . 8 |
28 | eleq2w 2219 | . . . . . . . . . 10 | |
29 | eqeq2 2167 | . . . . . . . . . 10 | |
30 | eleq1w 2218 | . . . . . . . . . 10 | |
31 | 28, 29, 30 | 3orbi123d 1293 | . . . . . . . . 9 |
32 | 31 | cbvralv 2680 | . . . . . . . 8 |
33 | 27, 32 | sylib 121 | . . . . . . 7 |
34 | 11, 12, 14, 16, 33 | exmidontriimlem3 7159 | . . . . . 6 |
35 | 34 | exp31 362 | . . . . 5 |
36 | 9, 35 | tfis2 4545 | . . . 4 |
37 | 36 | impcom 124 | . . 3 |
38 | 37 | ralrimiva 2530 | . 2 |
39 | exmidontriimlem4.b | . 2 | |
40 | 4, 38, 39 | rspcdva 2821 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3o 962 wceq 1335 wcel 2128 wral 2435 EXMIDwem 4156 con0 4324 |
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-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-nul 4091 ax-pow 4136 ax-setind 4497 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-dif 3104 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-uni 3774 df-tr 4064 df-exmid 4157 df-iord 4327 df-on 4329 |
This theorem is referenced by: exmidontriim 7161 |
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