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Theorem braba 5096
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 5093 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑅𝐵𝜓))
61, 2, 5mp2an 710 1 (𝐴𝑅𝐵𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1596  wcel 2103  Vcvv 3304   class class class wbr 4760  {copab 4820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-br 4761  df-opab 4821
This theorem is referenced by:  frgpuplem  18306  2ndcctbss  21381  legov  25600  prtlem13  34574  wepwsolem  38031  fnwe2val  38038  sprsymrelf  42172
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