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Theorem eloprabg 7255
Description: The law of concretion for operation class abstraction. Compare elopab 5401. (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 1431 . 2 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜃))
54eloprabga 7254 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  w3a 1084   = wceq 1538  wcel 2115  cop 4556  {coprab 7150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-oprab 7153
This theorem is referenced by:  ov  7287  ovg  7307  brbtwn  26696  isnvlem  28396  isphg  28603  fvtransport  33550  brcolinear2  33576  colineardim1  33579  fvray  33659  fvline  33662
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