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Mirrors > Home > MPE Home > Th. List > ovanraleqv | Structured version Visualization version GIF version |
Description: Equality theorem for a conjunction with an operation values within a restricted universal quantification. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 13-Aug-2022.) |
Ref | Expression |
---|---|
ovanraleqv.1 | โข (๐ต = ๐ โ (๐ โ ๐)) |
Ref | Expression |
---|---|
ovanraleqv | โข (๐ต = ๐ โ (โ๐ฅ โ ๐ (๐ โง (๐ด ยท ๐ต) = ๐ถ) โ โ๐ฅ โ ๐ (๐ โง (๐ด ยท ๐) = ๐ถ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovanraleqv.1 | . . 3 โข (๐ต = ๐ โ (๐ โ ๐)) | |
2 | oveq2 7419 | . . . 4 โข (๐ต = ๐ โ (๐ด ยท ๐ต) = (๐ด ยท ๐)) | |
3 | 2 | eqeq1d 2732 | . . 3 โข (๐ต = ๐ โ ((๐ด ยท ๐ต) = ๐ถ โ (๐ด ยท ๐) = ๐ถ)) |
4 | 1, 3 | anbi12d 629 | . 2 โข (๐ต = ๐ โ ((๐ โง (๐ด ยท ๐ต) = ๐ถ) โ (๐ โง (๐ด ยท ๐) = ๐ถ))) |
5 | 4 | ralbidv 3175 | 1 โข (๐ต = ๐ โ (โ๐ฅ โ ๐ (๐ โง (๐ด ยท ๐ต) = ๐ถ) โ โ๐ฅ โ ๐ (๐ โง (๐ด ยท ๐) = ๐ถ))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โ wb 205 โง wa 394 = wceq 1539 โwral 3059 (class class class)co 7411 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-ral 3060 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-iota 6494 df-fv 6550 df-ov 7414 |
This theorem is referenced by: mgmidmo 18585 ismgmid 18590 ismgmid2 18593 mgmidsssn0 18597 gsumvalx 18601 gsumress 18607 sgrpidmnd 18664 ismndd 18681 mnd1 18701 gsumvallem2 18751 mhmmnd 18983 ringurd 20079 opprring 20238 pzriprnglem7 21256 pzriprnglem13 21262 signsw0g 33865 signswmnd 33866 exidu1 37027 cmpidelt 37030 exidres 37049 exidresid 37050 isrngod 37069 rngoideu 37074 zlidlring 46914 2zrngamnd 46927 |
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