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Theorem exopxfr2 5845
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 5684 . . . . . . 7 (Rel 𝐴𝐴 ⊆ (V × V))
21biimpi 215 . . . . . 6 (Rel 𝐴𝐴 ⊆ (V × V))
32sseld 3982 . . . . 5 (Rel 𝐴 → (𝑥𝐴𝑥 ∈ (V × V)))
43adantrd 493 . . . 4 (Rel 𝐴 → ((𝑥𝐴𝜑) → 𝑥 ∈ (V × V)))
54pm4.71rd 564 . . 3 (Rel 𝐴 → ((𝑥𝐴𝜑) ↔ (𝑥 ∈ (V × V) ∧ (𝑥𝐴𝜑))))
65rexbidv2 3175 . 2 (Rel 𝐴 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥 ∈ (V × V)(𝑥𝐴𝜑)))
7 eleq1 2822 . . . 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 5844 . 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 205  wa 397   = wceq 1542  wex 1782  wcel 2107  wrex 3071  Vcvv 3475  wss 3949  cop 4635   × cxp 5675  Rel wrel 5682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-iun 5000  df-opab 5212  df-xp 5683  df-rel 5684
This theorem is referenced by:  dvhopellsm  39988
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