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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 7366 | . . . 4 โข (๐ต = ๐ โ (๐ด ยท ๐ต) = (๐ด ยท ๐)) | |
3 | 2 | eqeq1d 2739 | . . 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 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 397 = wceq 1542 โwral 3065 (class class class)co 7358 |
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 2708 |
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 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-iota 6449 df-fv 6505 df-ov 7361 |
This theorem is referenced by: mgmidmo 18516 ismgmid 18521 ismgmid2 18524 mgmidsssn0 18528 gsumvalx 18532 gsumress 18538 sgrpidmnd 18562 ismndd 18579 mnd1 18598 gsumvallem2 18645 mhmmnd 18870 rngurd 32068 signsw0g 33171 signswmnd 33172 exidu1 36318 cmpidelt 36321 exidres 36340 exidresid 36341 isrngod 36360 rngoideu 36365 zlidlring 46233 2zrngamnd 46246 |
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