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Theorem brab 4055
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 4052 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑅𝐵𝜒))
71, 2, 6mp2an 417 1 (𝐴𝑅𝐵𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1285  wcel 1434  Vcvv 2610   class class class wbr 3805  {copab 3858
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860
This theorem is referenced by:  dftpos4  5932  enq0sym  6736  enq0ref  6737  enq0tr  6738  shftfn  9913
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