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Theorem opabm 4276
Description: Inhabited ordered pair class abstraction. (Contributed by Jim Kingdon, 29-Sep-2018.)
Assertion
Ref Expression
opabm (∃𝑧 𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦𝜑)
Distinct variable groups:   𝜑,𝑧   𝑥,𝑧   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem opabm
StepHypRef Expression
1 elopab 4254 . . 3 (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
21exbii 1605 . 2 (∃𝑧 𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑧𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
3 exrot3 1690 . 2 (∃𝑧𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑦𝑧(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
4 vex 2740 . . . . . 6 𝑥 ∈ V
5 vex 2740 . . . . . 6 𝑦 ∈ V
64, 5opex 4225 . . . . 5 𝑥, 𝑦⟩ ∈ V
76isseti 2745 . . . 4 𝑧 𝑧 = ⟨𝑥, 𝑦
8 19.41v 1902 . . . 4 (∃𝑧(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (∃𝑧 𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
97, 8mpbiran 940 . . 3 (∃𝑧(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜑)
1092exbii 1606 . 2 (∃𝑥𝑦𝑧(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑦𝜑)
112, 3, 103bitri 206 1 (∃𝑧 𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦𝜑)
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105   = wceq 1353  wex 1492  wcel 2148  cop 3594  {copab 4060
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-opab 4062
This theorem is referenced by: (None)
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