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Mirrors > Home > ILE Home > Th. List > exmidontriimlem4 | Unicode version |
Description: Lemma for exmidontriim 7202. 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 2234 | . . 3 | |
2 | eqeq2 2180 | . . 3 | |
3 | eleq1 2233 | . . 3 | |
4 | 1, 2, 3 | 3orbi123d 1306 | . 2 |
5 | eleq2w 2232 | . . . . . . 7 | |
6 | eqeq2 2180 | . . . . . . 7 | |
7 | eleq1w 2231 | . . . . . . 7 | |
8 | 5, 6, 7 | 3orbi123d 1306 | . . . . . 6 |
9 | 8 | imbi2d 229 | . . . . 5 |
10 | exmidontriimlem4.a | . . . . . . . 8 | |
11 | 10 | adantl 275 | . . . . . . 7 |
12 | simpll 524 | . . . . . . 7 | |
13 | exmidontriimlem4.em | . . . . . . . 8 EXMID | |
14 | 13 | adantl 275 | . . . . . . 7 EXMID |
15 | exmidontriimlem4.h | . . . . . . . 8 | |
16 | 15 | adantl 275 | . . . . . . 7 |
17 | simplr 525 | . . . . . . . . . 10 | |
18 | eleq2w 2232 | . . . . . . . . . . . . 13 | |
19 | eqeq2 2180 | . . . . . . . . . . . . 13 | |
20 | eleq1w 2231 | . . . . . . . . . . . . 13 | |
21 | 18, 19, 20 | 3orbi123d 1306 | . . . . . . . . . . . 12 |
22 | 21 | imbi2d 229 | . . . . . . . . . . 11 |
23 | simpllr 529 | . . . . . . . . . . 11 | |
24 | simpr 109 | . . . . . . . . . . 11 | |
25 | 22, 23, 24 | rspcdva 2839 | . . . . . . . . . 10 |
26 | 17, 25 | mpd 13 | . . . . . . . . 9 |
27 | 26 | ralrimiva 2543 | . . . . . . . 8 |
28 | eleq2w 2232 | . . . . . . . . . 10 | |
29 | eqeq2 2180 | . . . . . . . . . 10 | |
30 | eleq1w 2231 | . . . . . . . . . 10 | |
31 | 28, 29, 30 | 3orbi123d 1306 | . . . . . . . . 9 |
32 | 31 | cbvralv 2696 | . . . . . . . 8 |
33 | 27, 32 | sylib 121 | . . . . . . 7 |
34 | 11, 12, 14, 16, 33 | exmidontriimlem3 7200 | . . . . . 6 |
35 | 34 | exp31 362 | . . . . 5 |
36 | 9, 35 | tfis2 4569 | . . . 4 |
37 | 36 | impcom 124 | . . 3 |
38 | 37 | ralrimiva 2543 | . 2 |
39 | exmidontriimlem4.b | . 2 | |
40 | 4, 38, 39 | rspcdva 2839 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3o 972 wceq 1348 wcel 2141 wral 2448 EXMIDwem 4180 con0 4348 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-setind 4521 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-uni 3797 df-tr 4088 df-exmid 4181 df-iord 4351 df-on 4353 |
This theorem is referenced by: exmidontriim 7202 |
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