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Theorem exopxfr 5836
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 5835 . 2 (∃𝑥 ∈ (V × V)𝜑 ↔ ∃𝑦 ∈ V ∃𝑧 ∈ V 𝜓)
3 rexv 3494 . 2 (∃𝑦 ∈ V ∃𝑧 ∈ V 𝜓 ↔ ∃𝑦𝑧 ∈ V 𝜓)
4 rexv 3494 . . 3 (∃𝑧 ∈ V 𝜓 ↔ ∃𝑧𝜓)
54exbii 1842 . 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 1533  wex 1773  wrex 3064  Vcvv 3468  cop 4629   × cxp 5667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-iun 4992  df-opab 5204  df-xp 5675  df-rel 5676
This theorem is referenced by:  exopxfr2  5837
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