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Theorem brabga 4979
 Description: The law of concretion for a binary relation. (Contributed by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
opelopabga.1 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
brabga.2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Assertion
Ref Expression
brabga ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵𝜓))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜓,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem brabga
StepHypRef Expression
1 df-br 4645 . . 3 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2 brabga.2 . . . 4 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
32eleq2i 2691 . . 3 (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
41, 3bitri 264 . 2 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
5 opelopabga.1 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
65opelopabga 4978 . 2 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜓))
74, 6syl5bb 272 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1481   ∈ wcel 1988  ⟨cop 4174   class class class wbr 4644  {copab 4703 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-opab 4704 This theorem is referenced by:  braba  4982  brabg  4984  epelg  5020  brcog  5277  fmptco  6382  ofrfval  6890  isfsupp  8264  wemaplem1  8436  oemapval  8565  wemapwe  8579  fpwwe2lem2  9439  fpwwelem  9452  clim  14206  rlim  14207  vdwmc  15663  isstruct2  15848  brssc  16455  isfunc  16505  isfull  16551  isfth  16555  ipole  17139  eqgval  17624  frgpuplem  18166  dvdsr  18627  islindf  20132  ulmval  24115  hpgbr  25633  isausgr  26040  issubgr  26144  isrgr  26436  isrusgr  26438  istrlson  26584  upgrwlkdvspth  26616  ispthson  26619  isspthson  26620  erclwwlkseq  26912  erclwwlksneq  26924  hlimi  28015  isinftm  29709  metidv  29909  ismntoplly  30043  brae  30278  braew  30279  brfae  30285  climf  39654  climf2  39698  nelbr  41054  iscllaw  41590  iscomlaw  41591  isasslaw  41593  islininds  42000  lindepsnlininds  42006
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