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Theorem eloprabg 7543
Description: The law of concretion for operation class abstraction. Compare elopab 5537. (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 1433 . 2 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜃))
54eloprabga 7541 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1086   = wceq 1537  wcel 2106  cop 4637  {coprab 7432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-oprab 7435
This theorem is referenced by:  ov  7577  ovg  7598  brbtwn  28929  isnvlem  30639  isphg  30846  fvtransport  36014  brcolinear2  36040  colineardim1  36043  fvray  36123  fvline  36126
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