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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 (∃!𝑦𝑥 𝑦 = 𝐴 → ∃!𝑦𝑥 𝑦 = 𝐴)
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem eusv2i
StepHypRef Expression
1 nfeu1 1959 . . 3 𝑦∃!𝑦𝑥 𝑦 = 𝐴
2 nfcvd 2229 . . . . . 6 (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝑦)
3 eusvnf 4275 . . . . . 6 (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
42, 3nfeqd 2243 . . . . 5 (∃!𝑦𝑥 𝑦 = 𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
5 nf2 1603 . . . . 5 (Ⅎ𝑥 𝑦 = 𝐴 ↔ (∃𝑥 𝑦 = 𝐴 → ∀𝑥 𝑦 = 𝐴))
64, 5sylib 120 . . . 4 (∃!𝑦𝑥 𝑦 = 𝐴 → (∃𝑥 𝑦 = 𝐴 → ∀𝑥 𝑦 = 𝐴))
7 19.2 1574 . . . 4 (∀𝑥 𝑦 = 𝐴 → ∃𝑥 𝑦 = 𝐴)
86, 7impbid1 140 . . 3 (∃!𝑦𝑥 𝑦 = 𝐴 → (∃𝑥 𝑦 = 𝐴 ↔ ∀𝑥 𝑦 = 𝐴))
91, 8eubid 1955 . 2 (∃!𝑦𝑥 𝑦 = 𝐴 → (∃!𝑦𝑥 𝑦 = 𝐴 ↔ ∃!𝑦𝑥 𝑦 = 𝐴))
109ibir 175 1 (∃!𝑦𝑥 𝑦 = 𝐴 → ∃!𝑦𝑥 𝑦 = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1287   = wceq 1289  wnf 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
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