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Theorem exopxfr2 5708
Description: Transfer ordered-pair existence from/to single variable existence. (Contributed by NM, 26-Feb-2014.)
Hypothesis
Ref Expression
exopxfr2.1 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))
Assertion
Ref Expression
exopxfr2 (Rel 𝐴 → (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝑧(⟨𝑦, 𝑧⟩ ∈ 𝐴𝜓)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝜑,𝑦,𝑧   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦,𝑧)

Proof of Theorem exopxfr2
StepHypRef Expression
1 df-rel 5555 . . . . . . 7 (Rel 𝐴𝐴 ⊆ (V × V))
21biimpi 217 . . . . . 6 (Rel 𝐴𝐴 ⊆ (V × V))
32sseld 3963 . . . . 5 (Rel 𝐴 → (𝑥𝐴𝑥 ∈ (V × V)))
43adantrd 492 . . . 4 (Rel 𝐴 → ((𝑥𝐴𝜑) → 𝑥 ∈ (V × V)))
54pm4.71rd 563 . . 3 (Rel 𝐴 → ((𝑥𝐴𝜑) ↔ (𝑥 ∈ (V × V) ∧ (𝑥𝐴𝜑))))
65rexbidv2 3292 . 2 (Rel 𝐴 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥 ∈ (V × V)(𝑥𝐴𝜑)))
7 eleq1 2897 . . . 4 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥𝐴 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
8 exopxfr2.1 . . . 4 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))
97, 8anbi12d 630 . . 3 (𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑥𝐴𝜑) ↔ (⟨𝑦, 𝑧⟩ ∈ 𝐴𝜓)))
109exopxfr 5707 . 2 (∃𝑥 ∈ (V × V)(𝑥𝐴𝜑) ↔ ∃𝑦𝑧(⟨𝑦, 𝑧⟩ ∈ 𝐴𝜓))
116, 10syl6bb 288 1 (Rel 𝐴 → (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝑧(⟨𝑦, 𝑧⟩ ∈ 𝐴𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wex 1771  wcel 2105  wrex 3136  Vcvv 3492  wss 3933  cop 4563   × cxp 5546  Rel wrel 5553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-iun 4912  df-opab 5120  df-xp 5554  df-rel 5555
This theorem is referenced by:  dvhopellsm  38133
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