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Theorem brabg 5127
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 493 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜒))
4 brabg.5 . 2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
53, 4brabga 5122 1 ((𝐴𝐶𝐵𝐷) → (𝐴𝑅𝐵𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1630  wcel 2144   class class class wbr 4784  {copab 4844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-opab 4845
This theorem is referenced by:  brab  5131  ideqg  5412  opelcnvg  5440  f1owe  6745  brrpssg  7085  bren  8117  brdomg  8118  brwdom  8627  ltprord  10053  shftfib  14019  efgrelexlema  18368  isref  21532  istrkgld  25578  islnopp  25851  axcontlem5  26068  cmbr  28777  leopg  29315  cvbr  29475  mdbr  29487  dmdbr  29492  soseq  32085  sltval  32131  brsslt  32231  isfne  32665  brabg2  33835  isriscg  34108  brssr  34586  lcvbr  34823
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