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Theorem exopxfr2 5299
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 5150 . . . . . . 7 (Rel 𝐴𝐴 ⊆ (V × V))
21biimpi 206 . . . . . 6 (Rel 𝐴𝐴 ⊆ (V × V))
32sseld 3635 . . . . 5 (Rel 𝐴 → (𝑥𝐴𝑥 ∈ (V × V)))
43adantrd 483 . . . 4 (Rel 𝐴 → ((𝑥𝐴𝜑) → 𝑥 ∈ (V × V)))
54pm4.71rd 668 . . 3 (Rel 𝐴 → ((𝑥𝐴𝜑) ↔ (𝑥 ∈ (V × V) ∧ (𝑥𝐴𝜑))))
65rexbidv2 3077 . 2 (Rel 𝐴 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥 ∈ (V × V)(𝑥𝐴𝜑)))
7 eleq1 2718 . . . 4 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥𝐴 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
8 exopxfr2.1 . . . 4 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))
97, 8anbi12d 747 . . 3 (𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑥𝐴𝜑) ↔ (⟨𝑦, 𝑧⟩ ∈ 𝐴𝜓)))
109exopxfr 5298 . 2 (∃𝑥 ∈ (V × V)(𝑥𝐴𝜑) ↔ ∃𝑦𝑧(⟨𝑦, 𝑧⟩ ∈ 𝐴𝜓))
116, 10syl6bb 276 1 (Rel 𝐴 → (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝑧(⟨𝑦, 𝑧⟩ ∈ 𝐴𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  wrex 2942  Vcvv 3231  wss 3607  cop 4216   × cxp 5141  Rel wrel 5148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-iun 4554  df-opab 4746  df-xp 5149  df-rel 5150
This theorem is referenced by:  dvhopellsm  36723
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