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Theorem opelopab 4995
Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 16-May-1995.)
Hypotheses
Ref Expression
opelopab.1 𝐴 ∈ V
opelopab.2 𝐵 ∈ V
opelopab.3 (𝑥 = 𝐴 → (𝜑𝜓))
opelopab.4 (𝑦 = 𝐵 → (𝜓𝜒))
Assertion
Ref Expression
opelopab (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜒,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem opelopab
StepHypRef Expression
1 opelopab.1 . 2 𝐴 ∈ V
2 opelopab.2 . 2 𝐵 ∈ V
3 opelopab.3 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
4 opelopab.4 . . 3 (𝑦 = 𝐵 → (𝜓𝜒))
53, 4opelopabg 4991 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))
61, 2, 5mp2an 708 1 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1482  wcel 1989  Vcvv 3198  cop 4181  {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-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-opab 4711
This theorem is referenced by:  opabid2  5249  dfres2  5451  f1oiso  6598  elopabi  7228  xporderlem  7285  cnlnssadj  28923  areacirclem5  33484  dicopelval  36292  dih1dimatlem  36444  pellexlem3  37221  fsovrfovd  38129
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