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Theorem exopxfr 5421
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 5420 . 2 (∃𝑥 ∈ (V × V)𝜑 ↔ ∃𝑦 ∈ V ∃𝑧 ∈ V 𝜓)
3 rexv 3360 . 2 (∃𝑦 ∈ V ∃𝑧 ∈ V 𝜓 ↔ ∃𝑦𝑧 ∈ V 𝜓)
4 rexv 3360 . . 3 (∃𝑧 ∈ V 𝜓 ↔ ∃𝑧𝜓)
54exbii 1923 . 2 (∃𝑦𝑧 ∈ V 𝜓 ↔ ∃𝑦𝑧𝜓)
62, 3, 53bitri 286 1 (∃𝑥 ∈ (V × V)𝜑 ↔ ∃𝑦𝑧𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1632  wex 1853  wrex 3051  Vcvv 3340  cop 4327   × cxp 5264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-iun 4674  df-opab 4865  df-xp 5272  df-rel 5273
This theorem is referenced by:  exopxfr2  5422
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