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Theorem brabg 5548
Description: The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
opelopabg.1 (𝑥 = 𝐴 → (𝜑𝜓))
opelopabg.2 (𝑦 = 𝐵 → (𝜓𝜒))
brabg.5 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Assertion
Ref Expression
brabg ((𝐴𝐶𝐵𝐷) → (𝐴𝑅𝐵𝜒))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜒,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑅(𝑥,𝑦)

Proof of Theorem brabg
StepHypRef Expression
1 opelopabg.1 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
2 opelopabg.2 . . 3 (𝑦 = 𝐵 → (𝜓𝜒))
31, 2sylan9bb 509 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜒))
4 brabg.5 . 2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
53, 4brabga 5543 1 ((𝐴𝐶𝐵𝐷) → (𝐴𝑅𝐵𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105   class class class wbr 5147  {copab 5209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210
This theorem is referenced by:  brab  5552  ideqg  5864  brcnvg  5892  f1owe  7372  brrpssg  7743  soseq  8182  breng  8992  brdom2g  8994  brdomgOLD  8996  brwdom  9604  brttrcl  9750  ltprord  11067  shftfib  15107  efgrelexlema  19781  isref  23532  sltval  27706  brsslt  27844  lrrecval  27986  istrkgld  28481  islnopp  28761  axcontlem5  28997  cmbr  31612  leopg  32150  cvbr  32310  mdbr  32322  dmdbr  32327  isfne  36321  brabg2  37703  isriscg  37970  brssr  38482  lcvbr  39002  bropabg  43312
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