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Theorem brab 5544
Description: The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.)
Hypotheses
Ref Expression
opelopab.1 𝐴 ∈ V
opelopab.2 𝐵 ∈ V
opelopab.3 (𝑥 = 𝐴 → (𝜑𝜓))
opelopab.4 (𝑦 = 𝐵 → (𝜓𝜒))
brab.5 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Assertion
Ref Expression
brab (𝐴𝑅𝐵𝜒)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜒,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝑅(𝑥,𝑦)

Proof of Theorem brab
StepHypRef Expression
1 opelopab.1 . 2 𝐴 ∈ V
2 opelopab.2 . 2 𝐵 ∈ V
3 opelopab.3 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
4 opelopab.4 . . 3 (𝑦 = 𝐵 → (𝜓𝜒))
5 brab.5 . . 3 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
63, 4, 5brabg 5540 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑅𝐵𝜒))
71, 2, 6mp2an 691 1 (𝐴𝑅𝐵𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wcel 2107  Vcvv 3475   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:  opbrop  5774  f1oweALT  7959  frxp  8112  fnwelem  8117  xpord2lem  8128  xpord3lem  8135  poseq  8144  dftpos4  8230  dfac3  10116  axdc2lem  10443  brdom7disj  10526  brdom6disj  10527  ordpipq  10937  ltresr  11135  shftfn  15020  2shfti  15027  ishpg  28010  brcgr  28158  ex-opab  29685  br8d  31839  br8  34726  br6  34727  br4  34728  dfbigcup2  34871  brsegle  35080  heiborlem2  36680
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