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Theorem euop2 5513
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 5465 . 2 𝐴, 𝑦⟩ ∈ V
2 euop2.1 . . 3 𝐴 ∈ V
32moop2 5503 . 2 ∃*𝑦 𝑥 = ⟨𝐴, 𝑦
41, 3euxfr2w 3717 1 (∃!𝑥𝑦(𝑥 = ⟨𝐴, 𝑦⟩ ∧ 𝜑) ↔ ∃!𝑦𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  ∃!weu 2563  Vcvv 3475  cop 4635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636
This theorem is referenced by:  dfac5lem1  10118
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