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Theorem brabg 5540
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 511 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜒))
4 brabg.5 . 2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
53, 4brabga 5535 1 ((𝐴𝐶𝐵𝐷) → (𝐴𝑅𝐵𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107   class class class wbr 5149  {copab 5211
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  ax-sep 5300  ax-nul 5307  ax-pr 5428
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-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-br 5150  df-opab 5212
This theorem is referenced by:  brab  5544  ideqg  5852  brcnvg  5880  f1owe  7350  brrpssg  7715  soseq  8145  breng  8948  brenOLD  8950  brdom2g  8951  brdomgOLD  8953  brwdom  9562  brttrcl  9708  ltprord  11025  shftfib  15019  efgrelexlema  19617  isref  23013  sltval  27150  brsslt  27287  lrrecval  27423  istrkgld  27710  islnopp  27990  axcontlem5  28226  cmbr  30837  leopg  31375  cvbr  31535  mdbr  31547  dmdbr  31552  isfne  35224  brabg2  36585  isriscg  36852  brssr  37371  lcvbr  37891  bropabg  42073
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