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Theorem brabg2 37075
Description: Relation by a binary relation abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypotheses
Ref Expression
brabg2.1 (𝑥 = 𝐴 → (𝜑𝜓))
brabg2.2 (𝑦 = 𝐵 → (𝜓𝜒))
brabg2.3 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
brabg2.4 (𝜒𝐴𝐶)
Assertion
Ref Expression
brabg2 (𝐵𝐷 → (𝐴𝑅𝐵𝜒))
Distinct variable groups:   𝑥,𝐴,𝑦   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 brabg2
StepHypRef Expression
1 brabg2.3 . . . . 5 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
21relopabiv 5810 . . . 4 Rel 𝑅
32brrelex1i 5722 . . 3 (𝐴𝑅𝐵𝐴 ∈ V)
4 brabg2.1 . . . . . . 7 (𝑥 = 𝐴 → (𝜑𝜓))
5 brabg2.2 . . . . . . 7 (𝑦 = 𝐵 → (𝜓𝜒))
64, 5, 1brabg 5529 . . . . . 6 ((𝐴 ∈ V ∧ 𝐵𝐷) → (𝐴𝑅𝐵𝜒))
76biimpd 228 . . . . 5 ((𝐴 ∈ V ∧ 𝐵𝐷) → (𝐴𝑅𝐵𝜒))
87ex 412 . . . 4 (𝐴 ∈ V → (𝐵𝐷 → (𝐴𝑅𝐵𝜒)))
98com3l 89 . . 3 (𝐵𝐷 → (𝐴𝑅𝐵 → (𝐴 ∈ V → 𝜒)))
103, 9mpdi 45 . 2 (𝐵𝐷 → (𝐴𝑅𝐵𝜒))
11 brabg2.4 . . 3 (𝜒𝐴𝐶)
124, 5, 1brabg 5529 . . . . 5 ((𝐴𝐶𝐵𝐷) → (𝐴𝑅𝐵𝜒))
1312exbiri 808 . . . 4 (𝐴𝐶 → (𝐵𝐷 → (𝜒𝐴𝑅𝐵)))
1413com3l 89 . . 3 (𝐵𝐷 → (𝜒 → (𝐴𝐶𝐴𝑅𝐵)))
1511, 14mpdi 45 . 2 (𝐵𝐷 → (𝜒𝐴𝑅𝐵))
1610, 15impbid 211 1 (𝐵𝐷 → (𝐴𝑅𝐵𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  Vcvv 3466   class class class wbr 5138  {copab 5200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-xp 5672  df-rel 5673
This theorem is referenced by: (None)
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