Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > sbthlemi9 | Unicode version |
Description: Lemma for isbth 6823. (Contributed by NM, 28-Mar-1998.) |
Ref | Expression |
---|---|
sbthlem.1 | |
sbthlem.2 | |
sbthlem.3 |
Ref | Expression |
---|---|
sbthlemi9 | EXMID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 967 | . . . . . . . . . 10 EXMID | |
2 | df-f1 5098 | . . . . . . . . . 10 | |
3 | 1, 2 | sylib 121 | . . . . . . . . 9 EXMID |
4 | 3 | simpld 111 | . . . . . . . 8 EXMID |
5 | df-f 5097 | . . . . . . . 8 | |
6 | 4, 5 | sylib 121 | . . . . . . 7 EXMID |
7 | 6 | simpld 111 | . . . . . 6 EXMID |
8 | df-fn 5096 | . . . . . 6 | |
9 | 7, 8 | sylib 121 | . . . . 5 EXMID |
10 | 9 | simpld 111 | . . . 4 EXMID |
11 | simp3 968 | . . . . . 6 EXMID | |
12 | df-f1 5098 | . . . . . 6 | |
13 | 11, 12 | sylib 121 | . . . . 5 EXMID |
14 | 13 | simprd 113 | . . . 4 EXMID |
15 | sbthlem.1 | . . . . 5 | |
16 | sbthlem.2 | . . . . 5 | |
17 | sbthlem.3 | . . . . 5 | |
18 | 15, 16, 17 | sbthlem7 6819 | . . . 4 |
19 | 10, 14, 18 | syl2anc 408 | . . 3 EXMID |
20 | simp1 966 | . . . 4 EXMID EXMID | |
21 | 9 | simprd 113 | . . . 4 EXMID |
22 | 13 | simpld 111 | . . . . . 6 EXMID |
23 | df-f 5097 | . . . . . 6 | |
24 | 22, 23 | sylib 121 | . . . . 5 EXMID |
25 | 24 | simprd 113 | . . . 4 EXMID |
26 | 15, 16, 17 | sbthlemi5 6817 | . . . 4 EXMID |
27 | 20, 21, 25, 26 | syl12anc 1199 | . . 3 EXMID |
28 | df-fn 5096 | . . 3 | |
29 | 19, 27, 28 | sylanbrc 413 | . 2 EXMID |
30 | 3 | simprd 113 | . . . 4 EXMID |
31 | 24 | simpld 111 | . . . . . 6 EXMID |
32 | df-fn 5096 | . . . . . 6 | |
33 | 31, 32 | sylib 121 | . . . . 5 EXMID |
34 | 33, 25 | jca 304 | . . . 4 EXMID |
35 | 15, 16, 17 | sbthlemi8 6820 | . . . 4 EXMID |
36 | 20, 30, 34, 14, 35 | syl22anc 1202 | . . 3 EXMID |
37 | 6 | simprd 113 | . . . 4 EXMID |
38 | 33 | simprd 113 | . . . . 5 EXMID |
39 | 38, 25 | jca 304 | . . . 4 EXMID |
40 | df-rn 4520 | . . . . 5 | |
41 | 15, 16, 17 | sbthlemi6 6818 | . . . . 5 EXMID |
42 | 40, 41 | syl5eqr 2164 | . . . 4 EXMID |
43 | 20, 37, 39, 14, 42 | syl22anc 1202 | . . 3 EXMID |
44 | df-fn 5096 | . . 3 | |
45 | 36, 43, 44 | sylanbrc 413 | . 2 EXMID |
46 | dff1o4 5343 | . 2 | |
47 | 29, 45, 46 | sylanbrc 413 | 1 EXMID |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 947 wceq 1316 wcel 1465 cab 2103 cvv 2660 cdif 3038 cun 3039 wss 3041 cuni 3706 EXMIDwem 4088 ccnv 4508 cdm 4509 crn 4510 cres 4511 cima 4512 wfun 5087 wfn 5088 wf 5089 wf1 5090 wf1o 5092 |
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 588 ax-in2 589 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-nul 4024 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-stab 801 df-dc 805 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-rab 2402 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-exmid 4089 df-id 4185 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-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 |
This theorem is referenced by: sbthlemi10 6822 |
Copyright terms: Public domain | W3C validator |