Theorem euop2 5458

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	⊢ (∃!𝑥∃𝑦(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

Step	Hyp	Ref	Expression
1		opex 5410	. 2 ⊢ ⟨𝐴, 𝑦⟩ ∈ V
2		euop2.1	. . 3 ⊢ 𝐴 ∈ V
3	2	moop2 5448	. 2 ⊢ ∃*𝑦 𝑥 = ⟨𝐴, 𝑦⟩
4	1, 3	euxfr2w 3676	1 ⊢ (∃!𝑥∃𝑦(𝑥 = ⟨𝐴, 𝑦⟩ ∧ 𝜑) ↔ ∃!𝑦𝜑)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∃!weu 2566  Vcvv 3438  ⟨cop 4584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585

This theorem is referenced by:  dfac5lem1  10031