Theorem eusv2i 5330

Description: Two ways to express single-valuedness of a class expression 𝐴(𝑥). (Contributed by NM, 14-Oct-2010.) (Revised by Mario Carneiro, 18-Nov-2016.)

Assertion

Ref	Expression
eusv2i	⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → ∃!𝑦∃𝑥 𝑦 = 𝐴)

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem eusv2i

Step	Hyp	Ref	Expression
1		nfeu1 2582	. . 3 ⊢ Ⅎ𝑦∃!𝑦∀𝑥 𝑦 = 𝐴
2		nfcvd 2893	. . . . . 6 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → Ⅎ𝑥𝑦)
3		eusvnf 5328	. . . . . 6 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → Ⅎ𝑥𝐴)
4	2, 3	nfeqd 2903	. . . . 5 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
5	4	nfrd 1792	. . . 4 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → (∃𝑥 𝑦 = 𝐴 → ∀𝑥 𝑦 = 𝐴))
6		19.2 1977	. . . 4 ⊢ (∀𝑥 𝑦 = 𝐴 → ∃𝑥 𝑦 = 𝐴)
7	5, 6	impbid1 225	. . 3 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → (∃𝑥 𝑦 = 𝐴 ↔ ∀𝑥 𝑦 = 𝐴))
8	1, 7	eubid 2581	. 2 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → (∃!𝑦∃𝑥 𝑦 = 𝐴 ↔ ∃!𝑦∀𝑥 𝑦 = 𝐴))
9	8	ibir 268	1 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → ∃!𝑦∃𝑥 𝑦 = 𝐴)

Syntax hints:   → wi 4  ∀wal 1539   = wceq 1541  ∃wex 1780  ∃!weu 2562

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 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-nul 4282

This theorem is referenced by:  eusv2nf  5331