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Theorem braba 4157
Description: The law of concretion for a binary relation. (Contributed by NM, 19-Dec-2013.)
Hypotheses
Ref Expression
opelopaba.1 𝐴 ∈ V
opelopaba.2 𝐵 ∈ V
opelopaba.3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
braba.4 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Assertion
Ref Expression
braba (𝐴𝑅𝐵𝜓)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜓,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑅(𝑥,𝑦)

Proof of Theorem braba
StepHypRef Expression
1 opelopaba.1 . 2 𝐴 ∈ V
2 opelopaba.2 . 2 𝐵 ∈ V
3 opelopaba.3 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
4 braba.4 . . 3 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
53, 4brabga 4154 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑅𝐵𝜓))
61, 2, 5mp2an 420 1 (𝐴𝑅𝐵𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1314  wcel 1463  Vcvv 2658   class class class wbr 3897  {copab 3956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958
This theorem is referenced by: (None)
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