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Theorem el2xptp 7968
Description: A member of a nested Cartesian product is an ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
Assertion
Ref Expression
el2xptp (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑥𝐵𝑦𝐶𝑧𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝐷,𝑦,𝑧

Proof of Theorem el2xptp
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 elxp2 5658 . 2 (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑝 ∈ (𝐵 × 𝐶)∃𝑧𝐷 𝐴 = ⟨𝑝, 𝑧⟩)
2 opeq1 4831 . . . . 5 (𝑝 = ⟨𝑥, 𝑦⟩ → ⟨𝑝, 𝑧⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
32eqeq2d 2748 . . . 4 (𝑝 = ⟨𝑥, 𝑦⟩ → (𝐴 = ⟨𝑝, 𝑧⟩ ↔ 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
43rexbidv 3176 . . 3 (𝑝 = ⟨𝑥, 𝑦⟩ → (∃𝑧𝐷 𝐴 = ⟨𝑝, 𝑧⟩ ↔ ∃𝑧𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
54rexxp 5799 . 2 (∃𝑝 ∈ (𝐵 × 𝐶)∃𝑧𝐷 𝐴 = ⟨𝑝, 𝑧⟩ ↔ ∃𝑥𝐵𝑦𝐶𝑧𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
6 df-ot 4596 . . . . . . 7 𝑥, 𝑦, 𝑧⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧
76eqcomi 2746 . . . . . 6 ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨𝑥, 𝑦, 𝑧
87eqeq2i 2750 . . . . 5 (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
98rexbii 3098 . . . 4 (∃𝑧𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ ∃𝑧𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
109rexbii 3098 . . 3 (∃𝑦𝐶𝑧𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ ∃𝑦𝐶𝑧𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
1110rexbii 3098 . 2 (∃𝑥𝐵𝑦𝐶𝑧𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ ∃𝑥𝐵𝑦𝐶𝑧𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
121, 5, 113bitri 297 1 (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑥𝐵𝑦𝐶𝑧𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1542  wcel 2107  wrex 3074  cop 4593  cotp 4595   × cxp 5632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-ot 4596  df-iun 4957  df-opab 5169  df-xp 5640  df-rel 5641
This theorem is referenced by:  el2xpss  7970  ralxp3f  8070  frpoins3xp3g  8074  poxp3  8083  xpord3pred  8085  sexp3  8086
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