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Theorem elxp2OLD 5093
Description: Obsolete proof of elxp2 5092 as of 13-Aug-2021. (Contributed by NM, 23-Feb-2004.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
elxp2OLD (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝐵𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦

Proof of Theorem elxp2OLD
StepHypRef Expression
1 df-rex 2913 . . . 4 (∃𝑦𝐶 (𝑥𝐵𝐴 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑦(𝑦𝐶 ∧ (𝑥𝐵𝐴 = ⟨𝑥, 𝑦⟩)))
2 r19.42v 3084 . . . 4 (∃𝑦𝐶 (𝑥𝐵𝐴 = ⟨𝑥, 𝑦⟩) ↔ (𝑥𝐵 ∧ ∃𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩))
3 an13 839 . . . . 5 ((𝑦𝐶 ∧ (𝑥𝐵𝐴 = ⟨𝑥, 𝑦⟩)) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
43exbii 1771 . . . 4 (∃𝑦(𝑦𝐶 ∧ (𝑥𝐵𝐴 = ⟨𝑥, 𝑦⟩)) ↔ ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
51, 2, 43bitr3i 290 . . 3 ((𝑥𝐵 ∧ ∃𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
65exbii 1771 . 2 (∃𝑥(𝑥𝐵 ∧ ∃𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
7 df-rex 2913 . 2 (∃𝑥𝐵𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥(𝑥𝐵 ∧ ∃𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩))
8 elxp 5091 . 2 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
96, 7, 83bitr4ri 293 1 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝐵𝑦𝐶 𝐴 = ⟨𝑥, 𝑦⟩)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  wrex 2908  cop 4154   × cxp 5072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2913  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-opab 4674  df-xp 5080
This theorem is referenced by: (None)
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