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Theorem brabg2 33990
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 5447 . . . 4 Rel 𝑅
32brrelex1i 5361 . . 3 (𝐴𝑅𝐵𝐴 ∈ V)
4 brabg2.1 . . . . . . 7 (𝑥 = 𝐴 → (𝜑𝜓))
5 brabg2.2 . . . . . . 7 (𝑦 = 𝐵 → (𝜓𝜒))
64, 5, 1brabg 5188 . . . . . 6 ((𝐴 ∈ V ∧ 𝐵𝐷) → (𝐴𝑅𝐵𝜒))
76biimpd 221 . . . . 5 ((𝐴 ∈ V ∧ 𝐵𝐷) → (𝐴𝑅𝐵𝜒))
87ex 402 . . . 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 5188 . . . . 5 ((𝐴𝐶𝐵𝐷) → (𝐴𝑅𝐵𝜒))
1312exbiri 846 . . . 4 (𝐴𝐶 → (𝐵𝐷 → (𝜒𝐴𝑅𝐵)))
1413com3l 89 . . 3 (𝐵𝐷 → (𝜒 → (𝐴𝐶𝐴𝑅𝐵)))
1511, 14mpdi 45 . 2 (𝐵𝐷 → (𝜒𝐴𝑅𝐵))
1610, 15impbid 204 1 (𝐵𝐷 → (𝐴𝑅𝐵𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  Vcvv 3383   class class class wbr 4841  {copab 4903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-br 4842  df-opab 4904  df-xp 5316  df-rel 5317
This theorem is referenced by: (None)
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