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Theorem opelopab 4163
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 4160 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))
61, 2, 5mp2an 422 1 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1316  wcel 1465  Vcvv 2660  cop 3500  {copab 3958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-opab 3960
This theorem is referenced by:  opabid2  4640  dfres2  4841  xporderlem  6096  acfun  7031  ccfunen  7047
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