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Theorem eusv2 4453
Description: Two ways to express single-valuedness of a class expression 𝐴(𝑥). (Contributed by NM, 15-Oct-2010.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
Hypothesis
Ref Expression
eusv2.1 𝐴 ∈ V
Assertion
Ref Expression
eusv2 (∃!𝑦𝑥 𝑦 = 𝐴 ↔ ∃!𝑦𝑥 𝑦 = 𝐴)
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem eusv2
StepHypRef Expression
1 eusv2.1 . . 3 𝐴 ∈ V
21eusv2nf 4452 . 2 (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
3 eusvnfb 4450 . . 3 (∃!𝑦𝑥 𝑦 = 𝐴 ↔ (𝑥𝐴𝐴 ∈ V))
41, 3mpbiran2 941 . 2 (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
52, 4bitr4i 187 1 (∃!𝑦𝑥 𝑦 = 𝐴 ↔ ∃!𝑦𝑥 𝑦 = 𝐴)
Colors of variables: wff set class
Syntax hints:  wb 105  wal 1351   = wceq 1353  wex 1492  ∃!weu 2026  wcel 2148  wnfc 2306  Vcvv 2737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-sn 3597  df-pr 3598  df-uni 3808
This theorem is referenced by: (None)
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