Theorem eusv2i 5400

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 2586	. . 3 ⊢ Ⅎ𝑦∃!𝑦∀𝑥 𝑦 = 𝐴
2		nfcvd 2904	. . . . . 6 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → Ⅎ𝑥𝑦)
3		eusvnf 5398	. . . . . 6 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → Ⅎ𝑥𝐴)
4	2, 3	nfeqd 2914	. . . . 5 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
5	4	nfrd 1788	. . . 4 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → (∃𝑥 𝑦 = 𝐴 → ∀𝑥 𝑦 = 𝐴))
6		19.2 1974	. . . 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 2585	. 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 1535   = wceq 1537  ∃wex 1776  ∃!weu 2566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-nul 4340

This theorem is referenced by:  eusv2nf  5401