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Theorem el2xptp 7446
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 5336 . 2 (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑝 ∈ (𝐵 × 𝐶)∃𝑧𝐷 𝐴 = ⟨𝑝, 𝑧⟩)
2 opeq1 4593 . . . . 5 (𝑝 = ⟨𝑥, 𝑦⟩ → ⟨𝑝, 𝑧⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
32eqeq2d 2809 . . . 4 (𝑝 = ⟨𝑥, 𝑦⟩ → (𝐴 = ⟨𝑝, 𝑧⟩ ↔ 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
43rexbidv 3233 . . 3 (𝑝 = ⟨𝑥, 𝑦⟩ → (∃𝑧𝐷 𝐴 = ⟨𝑝, 𝑧⟩ ↔ ∃𝑧𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
54rexxp 5468 . 2 (∃𝑝 ∈ (𝐵 × 𝐶)∃𝑧𝐷 𝐴 = ⟨𝑝, 𝑧⟩ ↔ ∃𝑥𝐵𝑦𝐶𝑧𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
6 df-ot 4377 . . . . . . 7 𝑥, 𝑦, 𝑧⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧
76eqcomi 2808 . . . . . 6 ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨𝑥, 𝑦, 𝑧
87eqeq2i 2811 . . . . 5 (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
98rexbii 3222 . . . 4 (∃𝑧𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ ∃𝑧𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
109rexbii 3222 . . 3 (∃𝑦𝐶𝑧𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ ∃𝑦𝐶𝑧𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
1110rexbii 3222 . 2 (∃𝑥𝐵𝑦𝐶𝑧𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ ∃𝑥𝐵𝑦𝐶𝑧𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
121, 5, 113bitri 289 1 (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑥𝐵𝑦𝐶𝑧𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
Colors of variables: wff setvar class
Syntax hints:  wb 198   = wceq 1653  wcel 2157  wrex 3090  cop 4374  cotp 4376   × cxp 5310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-ot 4377  df-iun 4712  df-opab 4906  df-xp 5318  df-rel 5319
This theorem is referenced by: (None)
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