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Theorem eusv2i 4893
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 2508 . . 3 𝑦∃!𝑦𝑥 𝑦 = 𝐴
2 nfcvd 2794 . . . . . 6 (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝑦)
3 eusvnf 4891 . . . . . 6 (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
42, 3nfeqd 2801 . . . . 5 (∃!𝑦𝑥 𝑦 = 𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
54nfrd 1757 . . . 4 (∃!𝑦𝑥 𝑦 = 𝐴 → (∃𝑥 𝑦 = 𝐴 → ∀𝑥 𝑦 = 𝐴))
6 19.2 1949 . . . 4 (∀𝑥 𝑦 = 𝐴 → ∃𝑥 𝑦 = 𝐴)
75, 6impbid1 215 . . 3 (∃!𝑦𝑥 𝑦 = 𝐴 → (∃𝑥 𝑦 = 𝐴 ↔ ∀𝑥 𝑦 = 𝐴))
81, 7eubid 2516 . 2 (∃!𝑦𝑥 𝑦 = 𝐴 → (∃!𝑦𝑥 𝑦 = 𝐴 ↔ ∃!𝑦𝑥 𝑦 = 𝐴))
98ibir 257 1 (∃!𝑦𝑥 𝑦 = 𝐴 → ∃!𝑦𝑥 𝑦 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1521   = wceq 1523  wex 1744  ∃!weu 2498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-nul 3949
This theorem is referenced by:  eusv2nf  4894
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