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Theorem exopxfr 5804
Description: Transfer ordered-pair existence from/to single variable existence. (Contributed by NM, 26-Feb-2014.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
exopxfr.1 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))
Assertion
Ref Expression
exopxfr (∃𝑥 ∈ (V × V)𝜑 ↔ ∃𝑦𝑧𝜓)
Distinct variable groups:   𝑦,𝑧,𝜑   𝜓,𝑥   𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦,𝑧)

Proof of Theorem exopxfr
StepHypRef Expression
1 exopxfr.1 . . 3 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))
21rexxp 5803 . 2 (∃𝑥 ∈ (V × V)𝜑 ↔ ∃𝑦 ∈ V ∃𝑧 ∈ V 𝜓)
3 rexv 3473 . 2 (∃𝑦 ∈ V ∃𝑧 ∈ V 𝜓 ↔ ∃𝑦𝑧 ∈ V 𝜓)
4 rexv 3473 . . 3 (∃𝑧 ∈ V 𝜓 ↔ ∃𝑧𝜓)
54exbii 1851 . 2 (∃𝑦𝑧 ∈ V 𝜓 ↔ ∃𝑦𝑧𝜓)
62, 3, 53bitri 297 1 (∃𝑥 ∈ (V × V)𝜑 ↔ ∃𝑦𝑧𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wex 1782  wrex 3074  Vcvv 3448  cop 4597   × cxp 5636
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 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
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 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-iun 4961  df-opab 5173  df-xp 5644  df-rel 5645
This theorem is referenced by:  exopxfr2  5805
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