Theorem eusv2i 4552

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 2090	. . 3 ⊢ Ⅎ𝑦∃!𝑦∀𝑥 𝑦 = 𝐴
2		nfcvd 2375	. . . . . 6 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → Ⅎ𝑥𝑦)
3		eusvnf 4550	. . . . . 6 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → Ⅎ𝑥𝐴)
4	2, 3	nfeqd 2389	. . . . 5 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
5		nf2 1716	. . . . 5 ⊢ (Ⅎ𝑥 𝑦 = 𝐴 ↔ (∃𝑥 𝑦 = 𝐴 → ∀𝑥 𝑦 = 𝐴))
6	4, 5	sylib 122	. . . 4 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → (∃𝑥 𝑦 = 𝐴 → ∀𝑥 𝑦 = 𝐴))
7		19.2 1686	. . . 4 ⊢ (∀𝑥 𝑦 = 𝐴 → ∃𝑥 𝑦 = 𝐴)
8	6, 7	impbid1 142	. . 3 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → (∃𝑥 𝑦 = 𝐴 ↔ ∀𝑥 𝑦 = 𝐴))
9	1, 8	eubid 2086	. 2 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → (∃!𝑦∃𝑥 𝑦 = 𝐴 ↔ ∃!𝑦∀𝑥 𝑦 = 𝐴))
10	9	ibir 177	1 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → ∃!𝑦∃𝑥 𝑦 = 𝐴)

Syntax hints:   → wi 4  ∀wal 1395   = wceq 1397  Ⅎwnf 1508  ∃wex 1540  ∃!weu 2079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-sbc 3032  df-csb 3128

This theorem is referenced by:  eusv2nf  4553