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Theorem exopxfr 5174
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 5173 . 2 (∃𝑥 ∈ (V × V)𝜑 ↔ ∃𝑦 ∈ V ∃𝑧 ∈ V 𝜓)
3 rexv 3192 . 2 (∃𝑦 ∈ V ∃𝑧 ∈ V 𝜓 ↔ ∃𝑦𝑧 ∈ V 𝜓)
4 rexv 3192 . . 3 (∃𝑧 ∈ V 𝜓 ↔ ∃𝑧𝜓)
54exbii 1763 . 2 (∃𝑦𝑧 ∈ V 𝜓 ↔ ∃𝑦𝑧𝜓)
62, 3, 53bitri 284 1 (∃𝑥 ∈ (V × V)𝜑 ↔ ∃𝑦𝑧𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194   = wceq 1474  wex 1694  wrex 2896  Vcvv 3172  cop 4130   × cxp 5025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-iun 4451  df-opab 4638  df-xp 5033  df-rel 5034
This theorem is referenced by:  exopxfr2  5175
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