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Theorem brabg2 33490
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}
21relopabi 5243 . . . 4 Rel 𝑅
32brrelexi 5156 . . 3 (𝐴𝑅𝐵𝐴 ∈ V)
4 brabg2.1 . . . . . . 7 (𝑥 = 𝐴 → (𝜑𝜓))
5 brabg2.2 . . . . . . 7 (𝑦 = 𝐵 → (𝜓𝜒))
64, 5, 1brabg 4992 . . . . . 6 ((𝐴 ∈ V ∧ 𝐵𝐷) → (𝐴𝑅𝐵𝜒))
76biimpd 219 . . . . 5 ((𝐴 ∈ V ∧ 𝐵𝐷) → (𝐴𝑅𝐵𝜒))
87ex 450 . . . 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 4992 . . . . 5 ((𝐴𝐶𝐵𝐷) → (𝐴𝑅𝐵𝜒))
1312exbiri 652 . . . 4 (𝐴𝐶 → (𝐵𝐷 → (𝜒𝐴𝑅𝐵)))
1413com3l 89 . . 3 (𝐵𝐷 → (𝜒 → (𝐴𝐶𝐴𝑅𝐵)))
1511, 14mpdi 45 . 2 (𝐵𝐷 → (𝜒𝐴𝑅𝐵))
1610, 15impbid 202 1 (𝐵𝐷 → (𝐴𝑅𝐵𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1482  wcel 1989  Vcvv 3198   class class class wbr 4651  {copab 4710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-br 4652  df-opab 4711  df-xp 5118  df-rel 5119
This theorem is referenced by: (None)
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