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Theorem eusv2i 4207
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 1953 . . 3 𝑦∃!𝑦𝑥 𝑦 = 𝐴
2 nfcvd 2221 . . . . . 6 (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝑦)
3 eusvnf 4205 . . . . . 6 (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
42, 3nfeqd 2234 . . . . 5 (∃!𝑦𝑥 𝑦 = 𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
5 nf2 1599 . . . . 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 1570 . . . 4 (∀𝑥 𝑦 = 𝐴 → ∃𝑥 𝑦 = 𝐴)
86, 7impbid1 140 . . 3 (∃!𝑦𝑥 𝑦 = 𝐴 → (∃𝑥 𝑦 = 𝐴 ↔ ∀𝑥 𝑦 = 𝐴))
91, 8eubid 1949 . 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 1283   = wceq 1285  wnf 1390  wex 1422  ∃!weu 1942
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-sbc 2817  df-csb 2910
This theorem is referenced by:  eusv2nf  4208
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