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Theorem brabg 4813
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 731 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜒))
4 brabg.5 . 2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
53, 4brabga 4808 1 ((𝐴𝐶𝐵𝐷) → (𝐴𝑅𝐵𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1938   class class class wbr 4481  {copab 4540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pr 4732
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-rab 2809  df-v 3079  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-br 4482  df-opab 4542
This theorem is referenced by:  brab  4817  ideqg  5087  opelcnvg  5116  f1owe  6379  brrpssg  6712  bren  7725  brdomg  7726  brwdom  8230  ltprord  9606  shftfib  13517  efgrelexlema  17891  isref  21023  istrkgld  25048  islnopp  25322  axcontlem5  25539  isfrgra  26256  cmbr  27616  leopg  28154  cvbr  28314  mdbr  28326  dmdbr  28331  soseq  30838  sltval  30880  isfne  31339  brabg2  32555  isriscg  32828  lcvbr  33201
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