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Theorem exopxfr 5187
 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 5186 . 2 (∃𝑥 ∈ (V × V)𝜑 ↔ ∃𝑦 ∈ V ∃𝑧 ∈ V 𝜓)
3 rexv 3193 . 2 (∃𝑦 ∈ V ∃𝑧 ∈ V 𝜓 ↔ ∃𝑦𝑧 ∈ V 𝜓)
4 rexv 3193 . . 3 (∃𝑧 ∈ V 𝜓 ↔ ∃𝑧𝜓)
54exbii 1764 . 2 (∃𝑦𝑧 ∈ V 𝜓 ↔ ∃𝑦𝑧𝜓)
62, 3, 53bitri 285 1 (∃𝑥 ∈ (V × V)𝜑 ↔ ∃𝑦𝑧𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475  ∃wex 1695  ∃wrex 2897  Vcvv 3173  ⟨cop 4131   × cxp 5036 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-iun 4457  df-opab 4644  df-xp 5044  df-rel 5045 This theorem is referenced by:  exopxfr2  5188
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