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Theorem opelopabg 4280
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 462 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜒))
43opelopabga 4275 1 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1363  wcel 2158  cop 3607  {copab 4075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-opab 4077
This theorem is referenced by:  opelopab  4283  fvopab3g  5602  fvopab3ig  5603  f1oiso  5840  ov  6007  ovg  6026  elopabi  6209  elinp  7486
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