Theorem euop2 5443

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 5396	. 2 ⊢ ⟨𝐴, 𝑦⟩ ∈ V
2		euop2.1	. . 3 ⊢ 𝐴 ∈ V
3	2	moop2 5433	. 2 ⊢ ∃*𝑦 𝑥 = ⟨𝐴, 𝑦⟩
4	1, 3	euxfr2w 3664	1 ⊢ (∃!𝑥∃𝑦(𝑥 = ⟨𝐴, 𝑦⟩ ∧ 𝜑) ↔ ∃!𝑦𝜑)

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1540  ∃wex 1780   ∈ wcel 2105  ∃!weu 2567  Vcvv 3441  ⟨cop 4575

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-sep 5236  ax-nul 5243  ax-pr 5365

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-rab 3405  df-v 3443  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576

This theorem is referenced by:  dfac5lem1  9949