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Theorem opelopabg 5538
Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
opelopabg.1 (𝑥 = 𝐴 → (𝜑𝜓))
opelopabg.2 (𝑦 = 𝐵 → (𝜓𝜒))
Assertion
Ref Expression
opelopabg ((𝐴𝑉𝐵𝑊) → (⟨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)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem opelopabg
StepHypRef Expression
1 opelopabg.1 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
2 opelopabg.2 . . 3 (𝑦 = 𝐵 → (𝜓𝜒))
31, 2sylan9bb 510 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜒))
43opelopabga 5533 1 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  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:  opelopab  5542  fvopab3g  6993  fvopab3ig  6994  ov  7554  ovg  7574  joindef  18331  meetdef  18345  isvclem  29868  adj1  31224  adjeq  31226  linedegen  35184  bj-finsumval0  36252  opelopab3  36672  isdivrngo  36904  iscom2  36949  inxprnres  37247  dihpN  40293  tfsconcatfv2  42172
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