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Theorem el2xptp 8040
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 5702 . 2 (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑝 ∈ (𝐵 × 𝐶)∃𝑧𝐷 𝐴 = ⟨𝑝, 𝑧⟩)
2 opeq1 4875 . . . . 5 (𝑝 = ⟨𝑥, 𝑦⟩ → ⟨𝑝, 𝑧⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
32eqeq2d 2736 . . . 4 (𝑝 = ⟨𝑥, 𝑦⟩ → (𝐴 = ⟨𝑝, 𝑧⟩ ↔ 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
43rexbidv 3168 . . 3 (𝑝 = ⟨𝑥, 𝑦⟩ → (∃𝑧𝐷 𝐴 = ⟨𝑝, 𝑧⟩ ↔ ∃𝑧𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
54rexxp 5845 . 2 (∃𝑝 ∈ (𝐵 × 𝐶)∃𝑧𝐷 𝐴 = ⟨𝑝, 𝑧⟩ ↔ ∃𝑥𝐵𝑦𝐶𝑧𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
6 df-ot 4639 . . . . . . 7 𝑥, 𝑦, 𝑧⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧
76eqcomi 2734 . . . . . 6 ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨𝑥, 𝑦, 𝑧
87eqeq2i 2738 . . . . 5 (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
98rexbii 3083 . . . 4 (∃𝑧𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ ∃𝑧𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
109rexbii 3083 . . 3 (∃𝑦𝐶𝑧𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ ∃𝑦𝐶𝑧𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
1110rexbii 3083 . 2 (∃𝑥𝐵𝑦𝐶𝑧𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ ∃𝑥𝐵𝑦𝐶𝑧𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
121, 5, 113bitri 296 1 (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑥𝐵𝑦𝐶𝑧𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1533  wcel 2098  wrex 3059  cop 4636  cotp 4638   × cxp 5676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-ot 4639  df-iun 4999  df-opab 5212  df-xp 5684  df-rel 5685
This theorem is referenced by:  el2xpss  8042  ralxp3f  8142  frpoins3xp3g  8146  poxp3  8155  xpord3pred  8157  sexp3  8158
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