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Theorem exopxfr2 5855
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 5692 . . . . . . 7 (Rel 𝐴𝐴 ⊆ (V × V))
21biimpi 216 . . . . . 6 (Rel 𝐴𝐴 ⊆ (V × V))
32sseld 3982 . . . . 5 (Rel 𝐴 → (𝑥𝐴𝑥 ∈ (V × V)))
43adantrd 491 . . . 4 (Rel 𝐴 → ((𝑥𝐴𝜑) → 𝑥 ∈ (V × V)))
54pm4.71rd 562 . . 3 (Rel 𝐴 → ((𝑥𝐴𝜑) ↔ (𝑥 ∈ (V × V) ∧ (𝑥𝐴𝜑))))
65rexbidv2 3175 . 2 (Rel 𝐴 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥 ∈ (V × V)(𝑥𝐴𝜑)))
7 eleq1 2829 . . . 4 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥𝐴 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
8 exopxfr2.1 . . . 4 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))
97, 8anbi12d 632 . . 3 (𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑥𝐴𝜑) ↔ (⟨𝑦, 𝑧⟩ ∈ 𝐴𝜓)))
109exopxfr 5854 . 2 (∃𝑥 ∈ (V × V)(𝑥𝐴𝜑) ↔ ∃𝑦𝑧(⟨𝑦, 𝑧⟩ ∈ 𝐴𝜓))
116, 10bitrdi 287 1 (Rel 𝐴 → (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝑧(⟨𝑦, 𝑧⟩ ∈ 𝐴𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  wrex 3070  Vcvv 3480  wss 3951  cop 4632   × cxp 5683  Rel wrel 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-iun 4993  df-opab 5206  df-xp 5691  df-rel 5692
This theorem is referenced by:  dvhopellsm  41119
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