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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 7417 | . . . 4 โข (๐ต = ๐ โ (๐ด ยท ๐ต) = (๐ด ยท ๐)) | |
3 | 2 | eqeq1d 2735 | . . 3 โข (๐ต = ๐ โ ((๐ด ยท ๐ต) = ๐ถ โ (๐ด ยท ๐) = ๐ถ)) |
4 | 1, 3 | anbi12d 632 | . 2 โข (๐ต = ๐ โ ((๐ โง (๐ด ยท ๐ต) = ๐ถ) โ (๐ โง (๐ด ยท ๐) = ๐ถ))) |
5 | 4 | ralbidv 3178 | 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 397 = wceq 1542 โwral 3062 (class class class)co 7409 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-iota 6496 df-fv 6552 df-ov 7412 |
This theorem is referenced by: mgmidmo 18579 ismgmid 18584 ismgmid2 18587 mgmidsssn0 18591 gsumvalx 18595 gsumress 18601 sgrpidmnd 18630 ismndd 18647 mnd1 18667 gsumvallem2 18715 mhmmnd 18947 ringurd 20008 signsw0g 33567 signswmnd 33568 exidu1 36724 cmpidelt 36727 exidres 36746 exidresid 36747 isrngod 36766 rngoideu 36771 pzriprnglem7 46811 pzriprnglem13 46817 zlidlring 46826 2zrngamnd 46839 |
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