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Theorem eloprabg 6646
 Description: The law of concretion for operation class abstraction. Compare elopab 4908. (Contributed by NM, 14-Sep-1999.) (Revised by David Abernethy, 19-Jun-2012.)
Hypotheses
Ref Expression
eloprabg.1 (𝑥 = 𝐴 → (𝜑𝜓))
eloprabg.2 (𝑦 = 𝐵 → (𝜓𝜒))
eloprabg.3 (𝑧 = 𝐶 → (𝜒𝜃))
Assertion
Ref Expression
eloprabg ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜃))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝜃,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)   𝜒(𝑥,𝑦,𝑧)   𝑉(𝑥,𝑦,𝑧)   𝑊(𝑥,𝑦,𝑧)   𝑋(𝑥,𝑦,𝑧)

Proof of Theorem eloprabg
StepHypRef Expression
1 eloprabg.1 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
2 eloprabg.2 . . 3 (𝑦 = 𝐵 → (𝜓𝜒))
3 eloprabg.3 . . 3 (𝑧 = 𝐶 → (𝜒𝜃))
41, 2, 3syl3an9b 1389 . 2 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜃))
54eloprabga 6645 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜃))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ⟨cop 4131  {coprab 6550 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-oprab 6553 This theorem is referenced by:  ov  6678  ovg  6697  brbtwn  25579  isnvlem  26849  isphg  27056  fvtransport  31309  brcolinear2  31335  colineardim1  31338  fvray  31418  fvline  31421
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