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Theorem opelopabga 5533
Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.)
Hypothesis
Ref Expression
opelopabga.1 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
Assertion
Ref Expression
opelopabga ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜓))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜓,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem opelopabga
StepHypRef Expression
1 elopab 5527 . 2 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
2 opelopabga.1 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
32copsex2g 5493 . 2 ((𝐴𝑉𝐵𝑊) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓))
41, 3bitrid 282 1 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wex 1781  wcel 2106  cop 4634  {copab 5210
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-opab 5211
This theorem is referenced by:  brabga  5534  opelopab2a  5535  opelopaba  5536  opelopabg  5538  rbropapd  5564  fmptsng  7165  canthwelem  10644  iswlk  28864  fmlafvel  34371  satfv0fvfmla0  34399  satfv1fvfmla1  34409  isrngo  36760  tfsconcatb0  42084  sprsymrelf1lem  46149  isupwlk  46504  uspgrsprfo  46516
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