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Theorem euop2 5003
Description: Transfer existential uniqueness to second member of an ordered pair. (Contributed by NM, 10-Apr-2004.)
Hypothesis
Ref Expression
euop2.1 𝐴 ∈ V
Assertion
Ref Expression
euop2 (∃!𝑥𝑦(𝑥 = ⟨𝐴, 𝑦⟩ ∧ 𝜑) ↔ ∃!𝑦𝜑)
Distinct variable groups:   𝜑,𝑥   𝑥,𝐴   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)

Proof of Theorem euop2
StepHypRef Expression
1 opex 4962 . 2 𝐴, 𝑦⟩ ∈ V
2 euop2.1 . . 3 𝐴 ∈ V
32moop2 4995 . 2 ∃*𝑦 𝑥 = ⟨𝐴, 𝑦
41, 3euxfr2 3424 1 (∃!𝑥𝑦(𝑥 = ⟨𝐴, 𝑦⟩ ∧ 𝜑) ↔ ∃!𝑦𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  ∃!weu 2498  Vcvv 3231  cop 4216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217
This theorem is referenced by:  dfac5lem1  8984
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