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Theorem bj-uniex2 10036
 Description: uniex2 4173 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-uniex2 𝑦 𝑦 = 𝑥
Distinct variable group:   𝑥,𝑦

Proof of Theorem bj-uniex2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 bdcuni 9996 . . . 4 BOUNDED 𝑥
21bdeli 9966 . . 3 BOUNDED 𝑧 𝑥
3 zfun 4171 . . . 4 𝑦𝑧(∃𝑦(𝑧𝑦𝑦𝑥) → 𝑧𝑦)
4 eluni 3583 . . . . . . 7 (𝑧 𝑥 ↔ ∃𝑦(𝑧𝑦𝑦𝑥))
54imbi1i 227 . . . . . 6 ((𝑧 𝑥𝑧𝑦) ↔ (∃𝑦(𝑧𝑦𝑦𝑥) → 𝑧𝑦))
65albii 1359 . . . . 5 (∀𝑧(𝑧 𝑥𝑧𝑦) ↔ ∀𝑧(∃𝑦(𝑧𝑦𝑦𝑥) → 𝑧𝑦))
76exbii 1496 . . . 4 (∃𝑦𝑧(𝑧 𝑥𝑧𝑦) ↔ ∃𝑦𝑧(∃𝑦(𝑧𝑦𝑦𝑥) → 𝑧𝑦))
83, 7mpbir 134 . . 3 𝑦𝑧(𝑧 𝑥𝑧𝑦)
92, 8bdbm1.3ii 10011 . 2 𝑦𝑧(𝑧𝑦𝑧 𝑥)
10 dfcleq 2034 . . 3 (𝑦 = 𝑥 ↔ ∀𝑧(𝑧𝑦𝑧 𝑥))
1110exbii 1496 . 2 (∃𝑦 𝑦 = 𝑥 ↔ ∃𝑦𝑧(𝑧𝑦𝑧 𝑥))
129, 11mpbir 134 1 𝑦 𝑦 = 𝑥
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1241   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ∪ cuni 3580 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-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-un 4170  ax-bd0 9933  ax-bdex 9939  ax-bdel 9941  ax-bdsb 9942  ax-bdsep 10004 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-uni 3581  df-bdc 9961 This theorem is referenced by:  bj-uniex  10037
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