Theorem euop2 5494

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 5444	. 2 ⊢ ⟨𝐴, 𝑦⟩ ∈ V
2		euop2.1	. . 3 ⊢ 𝐴 ∈ V
3	2	moop2 5484	. 2 ⊢ ∃*𝑦 𝑥 = ⟨𝐴, 𝑦⟩
4	1, 3	euxfr2w 3682	1 ⊢ (∃!𝑥∃𝑦(𝑥 = ⟨𝐴, 𝑦⟩ ∧ 𝜑) ↔ ∃!𝑦𝜑)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 400   = wceq 1569  ∃wex 1808   ∈ wcel 2142  ∃!weu 2595  Vcvv 3454  ⟨cop 4594

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595

This theorem is used by:  dfac5lem1  10114