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Theorem brcodir 4655
Description: Two ways of saying that two elements have an upper bound. (Contributed by Mario Carneiro, 3-Nov-2015.)
Assertion
Ref Expression
brcodir ((A 𝑉 B 𝑊) → (A(𝑅𝑅)Bz(A𝑅z B𝑅z)))
Distinct variable groups:   z,A   z,B   z,𝑅   z,𝑉   z,𝑊

Proof of Theorem brcodir
StepHypRef Expression
1 brcog 4445 . 2 ((A 𝑉 B 𝑊) → (A(𝑅𝑅)Bz(A𝑅z z𝑅B)))
2 vex 2554 . . . . . 6 z V
3 brcnvg 4459 . . . . . 6 ((z V B 𝑊) → (z𝑅BB𝑅z))
42, 3mpan 400 . . . . 5 (B 𝑊 → (z𝑅BB𝑅z))
54anbi2d 437 . . . 4 (B 𝑊 → ((A𝑅z z𝑅B) ↔ (A𝑅z B𝑅z)))
65adantl 262 . . 3 ((A 𝑉 B 𝑊) → ((A𝑅z z𝑅B) ↔ (A𝑅z B𝑅z)))
76exbidv 1703 . 2 ((A 𝑉 B 𝑊) → (z(A𝑅z z𝑅B) ↔ z(A𝑅z B𝑅z)))
81, 7bitrd 177 1 ((A 𝑉 B 𝑊) → (A(𝑅𝑅)Bz(A𝑅z B𝑅z)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wex 1378   wcel 1390  Vcvv 2551   class class class wbr 3755  ccnv 4287  ccom 4292
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-cnv 4296  df-co 4297
This theorem is referenced by:  codir  4656
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