Theorem euop2 5450

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 5402	. 2 ⊢ ⟨𝐴, 𝑦⟩ ∈ V
2		euop2.1	. . 3 ⊢ 𝐴 ∈ V
3	2	moop2 5440	. 2 ⊢ ∃*𝑦 𝑥 = ⟨𝐴, 𝑦⟩
4	1, 3	euxfr2w 3674	1 ⊢ (∃!𝑥∃𝑦(𝑥 = ⟨𝐴, 𝑦⟩ ∧ 𝜑) ↔ ∃!𝑦𝜑)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2111  ∃!weu 2563  Vcvv 3436  ⟨cop 4579

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580

This theorem is referenced by:  dfac5lem1  10014