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Theorem opelopab 5548
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 5544 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))
61, 2, 5mp2an 690 1 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1534  wcel 2099  Vcvv 3462  cop 4639  {copab 5215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-opab 5216
This theorem is referenced by:  opabid2  5834  dfres2  6050  f1oiso  7363  elopabi  8076  xporderlem  8141  cnlnssadj  32013  areacirclem5  37413  dicopelval  40876  dih1dimatlem  41028  pellexlem3  42488  fsovrfovd  43676
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