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Mirrors > Home > MPE Home > Th. List > hasheqf1oiOLD | Structured version Visualization version Unicode version |
Description: Obsolete version of hasheqf1oi 13140 as of 4-May-2021. (Contributed by Alexander van der Vekens, 25-Dec-2017.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
hasheqf1oiOLD |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hasheqf1o 13137 | . . . 4 | |
2 | 1 | biimprd 238 | . . 3 |
3 | 2 | a1d 25 | . 2 |
4 | fiinfnf1o 13138 | . . . 4 | |
5 | 4 | pm2.21d 118 | . . 3 |
6 | 5 | a1d 25 | . 2 |
7 | fiinfnf1o 13138 | . . . 4 | |
8 | 19.41v 1914 | . . . . . . 7 | |
9 | f1orel 6140 | . . . . . . . . . . . . 13 | |
10 | 9 | adantr 481 | . . . . . . . . . . . 12 |
11 | f1ocnvb 6150 | . . . . . . . . . . . 12 | |
12 | 10, 11 | syl 17 | . . . . . . . . . . 11 |
13 | f1of 6137 | . . . . . . . . . . . . . . 15 | |
14 | 13 | adantr 481 | . . . . . . . . . . . . . 14 |
15 | simprl 794 | . . . . . . . . . . . . . 14 | |
16 | simprr 796 | . . . . . . . . . . . . . 14 | |
17 | fex2 7121 | . . . . . . . . . . . . . 14 | |
18 | 14, 15, 16, 17 | syl3anc 1326 | . . . . . . . . . . . . 13 |
19 | cnvexg 7112 | . . . . . . . . . . . . 13 | |
20 | f1oeq1 6127 | . . . . . . . . . . . . . 14 | |
21 | 20 | spcegv 3294 | . . . . . . . . . . . . 13 |
22 | 18, 19, 21 | 3syl 18 | . . . . . . . . . . . 12 |
23 | pm2.24 121 | . . . . . . . . . . . 12 | |
24 | 22, 23 | syl6 35 | . . . . . . . . . . 11 |
25 | 12, 24 | sylbid 230 | . . . . . . . . . 10 |
26 | 25 | com12 32 | . . . . . . . . 9 |
27 | 26 | anabsi5 858 | . . . . . . . 8 |
28 | 27 | exlimiv 1858 | . . . . . . 7 |
29 | 8, 28 | sylbir 225 | . . . . . 6 |
30 | 29 | ex 450 | . . . . 5 |
31 | 30 | com13 88 | . . . 4 |
32 | 7, 31 | syl 17 | . . 3 |
33 | 32 | ancoms 469 | . 2 |
34 | hashinf 13122 | . . . . . . . . . 10 | |
35 | 34 | expcom 451 | . . . . . . . . 9 |
36 | 35 | adantr 481 | . . . . . . . 8 |
37 | 36 | com12 32 | . . . . . . 7 |
38 | 37 | adantr 481 | . . . . . 6 |
39 | 38 | impcom 446 | . . . . 5 |
40 | hashinf 13122 | . . . . . . . . . 10 | |
41 | 40 | expcom 451 | . . . . . . . . 9 |
42 | 41 | adantl 482 | . . . . . . . 8 |
43 | 42 | com12 32 | . . . . . . 7 |
44 | 43 | adantl 482 | . . . . . 6 |
45 | 44 | impcom 446 | . . . . 5 |
46 | 39, 45 | eqtr4d 2659 | . . . 4 |
47 | 46 | a1d 25 | . . 3 |
48 | 47 | ex 450 | . 2 |
49 | 3, 6, 33, 48 | 4cases 990 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wa 384 wceq 1483 wex 1704 wcel 1990 cvv 3200 ccnv 5113 wrel 5119 wf 5884 wf1o 5887 cfv 5888 cfn 7955 cpnf 10071 chash 13117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-hash 13118 |
This theorem is referenced by: hashf1rnOLD 13143 |
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