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Theorem eusv2i 4440
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 2030 . . 3 𝑦∃!𝑦𝑥 𝑦 = 𝐴
2 nfcvd 2313 . . . . . 6 (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝑦)
3 eusvnf 4438 . . . . . 6 (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
42, 3nfeqd 2327 . . . . 5 (∃!𝑦𝑥 𝑦 = 𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
5 nf2 1661 . . . . 5 (Ⅎ𝑥 𝑦 = 𝐴 ↔ (∃𝑥 𝑦 = 𝐴 → ∀𝑥 𝑦 = 𝐴))
64, 5sylib 121 . . . 4 (∃!𝑦𝑥 𝑦 = 𝐴 → (∃𝑥 𝑦 = 𝐴 → ∀𝑥 𝑦 = 𝐴))
7 19.2 1631 . . . 4 (∀𝑥 𝑦 = 𝐴 → ∃𝑥 𝑦 = 𝐴)
86, 7impbid1 141 . . 3 (∃!𝑦𝑥 𝑦 = 𝐴 → (∃𝑥 𝑦 = 𝐴 ↔ ∀𝑥 𝑦 = 𝐴))
91, 8eubid 2026 . 2 (∃!𝑦𝑥 𝑦 = 𝐴 → (∃!𝑦𝑥 𝑦 = 𝐴 ↔ ∃!𝑦𝑥 𝑦 = 𝐴))
109ibir 176 1 (∃!𝑦𝑥 𝑦 = 𝐴 → ∃!𝑦𝑥 𝑦 = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1346   = wceq 1348  wnf 1453  wex 1485  ∃!weu 2019
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956  df-csb 3050
This theorem is referenced by:  eusv2nf  4441
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