Theorem eusv2i 4277

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 1959	. . 3 ⊢ Ⅎ𝑦∃!𝑦∀𝑥 𝑦 = 𝐴
2		nfcvd 2229	. . . . . 6 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → Ⅎ𝑥𝑦)
3		eusvnf 4275	. . . . . 6 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → Ⅎ𝑥𝐴)
4	2, 3	nfeqd 2243	. . . . 5 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
5		nf2 1603	. . . . 5 ⊢ (Ⅎ𝑥 𝑦 = 𝐴 ↔ (∃𝑥 𝑦 = 𝐴 → ∀𝑥 𝑦 = 𝐴))
6	4, 5	sylib 120	. . . 4 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → (∃𝑥 𝑦 = 𝐴 → ∀𝑥 𝑦 = 𝐴))
7		19.2 1574	. . . 4 ⊢ (∀𝑥 𝑦 = 𝐴 → ∃𝑥 𝑦 = 𝐴)
8	6, 7	impbid1 140	. . 3 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → (∃𝑥 𝑦 = 𝐴 ↔ ∀𝑥 𝑦 = 𝐴))
9	1, 8	eubid 1955	. 2 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → (∃!𝑦∃𝑥 𝑦 = 𝐴 ↔ ∃!𝑦∀𝑥 𝑦 = 𝐴))
10	9	ibir 175	1 ⊢ (∃!𝑦∀𝑥 𝑦 = 𝐴 → ∃!𝑦∃𝑥 𝑦 = 𝐴)

Syntax hints:   → wi 4  ∀wal 1287   = wceq 1289  Ⅎwnf 1394  ∃wex 1426  ∃!weu 1948

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sbc 2841  df-csb 2934

This theorem is referenced by:  eusv2nf  4278