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Theorem ovanraleqv 7433
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.)
Hypothesis
Ref Expression
ovanraleqv.1 (๐ต = ๐‘‹ โ†’ (๐œ‘ โ†” ๐œ“))
Assertion
Ref Expression
ovanraleqv (๐ต = ๐‘‹ โ†’ (โˆ€๐‘ฅ โˆˆ ๐‘‰ (๐œ‘ โˆง (๐ด ยท ๐ต) = ๐ถ) โ†” โˆ€๐‘ฅ โˆˆ ๐‘‰ (๐œ“ โˆง (๐ด ยท ๐‘‹) = ๐ถ)))
Distinct variable groups:   ๐‘ฅ,๐ต   ๐‘ฅ,๐‘‹
Allowed substitution hints:   ๐œ‘(๐‘ฅ)   ๐œ“(๐‘ฅ)   ๐ด(๐‘ฅ)   ๐ถ(๐‘ฅ)   ยท (๐‘ฅ)   ๐‘‰(๐‘ฅ)

Proof of Theorem ovanraleqv
StepHypRef Expression
1 ovanraleqv.1 . . 3 (๐ต = ๐‘‹ โ†’ (๐œ‘ โ†” ๐œ“))
2 oveq2 7417 . . . 4 (๐ต = ๐‘‹ โ†’ (๐ด ยท ๐ต) = (๐ด ยท ๐‘‹))
32eqeq1d 2735 . . 3 (๐ต = ๐‘‹ โ†’ ((๐ด ยท ๐ต) = ๐ถ โ†” (๐ด ยท ๐‘‹) = ๐ถ))
41, 3anbi12d 632 . 2 (๐ต = ๐‘‹ โ†’ ((๐œ‘ โˆง (๐ด ยท ๐ต) = ๐ถ) โ†” (๐œ“ โˆง (๐ด ยท ๐‘‹) = ๐ถ)))
54ralbidv 3178 1 (๐ต = ๐‘‹ โ†’ (โˆ€๐‘ฅ โˆˆ ๐‘‰ (๐œ‘ โˆง (๐ด ยท ๐ต) = ๐ถ) โ†” โˆ€๐‘ฅ โˆˆ ๐‘‰ (๐œ“ โˆง (๐ด ยท ๐‘‹) = ๐ถ)))
Colors of variables: wff setvar class
Syntax hints:   โ†’ wi 4   โ†” wb 205   โˆง wa 397   = wceq 1542  โˆ€wral 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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