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Theorem brabg 4096
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 450 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜒))
4 brabg.5 . 2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
53, 4brabga 4091 1 ((𝐴𝐶𝐵𝐷) → (𝐴𝑅𝐵𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wcel 1438   class class class wbr 3845  {copab 3898
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900
This theorem is referenced by:  brab  4099  opbrop  4517  ideqg  4587  opelcnvg  4616  bren  6464  brdomg  6465  enq0breq  6995  ltresr  7376  ltxrlt  7552  apreap  8064  apreim  8080  shftfibg  10254  shftfib  10257  2shfti  10265
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