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Theorem eusvobj1 5499
 Description: Specify the same object in two ways when class 𝐵(𝑦) is single-valued. (Contributed by NM, 1-Nov-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
Hypothesis
Ref Expression
eusvobj1.1 𝐵 ∈ V
Assertion
Ref Expression
eusvobj1 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → (℩𝑥𝑦𝐴 𝑥 = 𝐵) = (℩𝑥𝑦𝐴 𝑥 = 𝐵))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐵(𝑦)

Proof of Theorem eusvobj1
StepHypRef Expression
1 nfeu1 1911 . . 3 𝑥∃!𝑥𝑦𝐴 𝑥 = 𝐵
2 eusvobj1.1 . . . 4 𝐵 ∈ V
32eusvobj2 5498 . . 3 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → (∃𝑦𝐴 𝑥 = 𝐵 ↔ ∀𝑦𝐴 𝑥 = 𝐵))
41, 3alrimi 1415 . 2 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∀𝑥(∃𝑦𝐴 𝑥 = 𝐵 ↔ ∀𝑦𝐴 𝑥 = 𝐵))
5 iotabi 4876 . 2 (∀𝑥(∃𝑦𝐴 𝑥 = 𝐵 ↔ ∀𝑦𝐴 𝑥 = 𝐵) → (℩𝑥𝑦𝐴 𝑥 = 𝐵) = (℩𝑥𝑦𝐴 𝑥 = 𝐵))
64, 5syl 14 1 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → (℩𝑥𝑦𝐴 𝑥 = 𝐵) = (℩𝑥𝑦𝐴 𝑥 = 𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241   = wceq 1243   ∈ wcel 1393  ∃!weu 1900  ∀wral 2306  ∃wrex 2307  Vcvv 2557  ℩cio 4865 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-csb 2853  df-sn 3381  df-uni 3581  df-iota 4867 This theorem is referenced by: (None)
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