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Theorem bj-uniex2 10850
Description: uniex2 4193 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 10810 . . . 4 BOUNDED 𝑥
21bdeli 10780 . . 3 BOUNDED 𝑧 𝑥
3 zfun 4191 . . . 4 𝑦𝑧(∃𝑦(𝑧𝑦𝑦𝑥) → 𝑧𝑦)
4 eluni 3606 . . . . . . 7 (𝑧 𝑥 ↔ ∃𝑦(𝑧𝑦𝑦𝑥))
54imbi1i 236 . . . . . 6 ((𝑧 𝑥𝑧𝑦) ↔ (∃𝑦(𝑧𝑦𝑦𝑥) → 𝑧𝑦))
65albii 1400 . . . . 5 (∀𝑧(𝑧 𝑥𝑧𝑦) ↔ ∀𝑧(∃𝑦(𝑧𝑦𝑦𝑥) → 𝑧𝑦))
76exbii 1537 . . . 4 (∃𝑦𝑧(𝑧 𝑥𝑧𝑦) ↔ ∃𝑦𝑧(∃𝑦(𝑧𝑦𝑦𝑥) → 𝑧𝑦))
83, 7mpbir 144 . . 3 𝑦𝑧(𝑧 𝑥𝑧𝑦)
92, 8bdbm1.3ii 10825 . 2 𝑦𝑧(𝑧𝑦𝑧 𝑥)
10 dfcleq 2076 . . 3 (𝑦 = 𝑥 ↔ ∀𝑧(𝑧𝑦𝑧 𝑥))
1110exbii 1537 . 2 (∃𝑦 𝑦 = 𝑥 ↔ ∃𝑦𝑧(𝑧𝑦𝑧 𝑥))
129, 11mpbir 144 1 𝑦 𝑦 = 𝑥
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1283   = wceq 1285  wex 1422  wcel 1434   cuni 3603
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-un 4190  ax-bd0 10747  ax-bdex 10753  ax-bdel 10755  ax-bdsb 10756  ax-bdsep 10818
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-v 2604  df-uni 3604  df-bdc 10775
This theorem is referenced by:  bj-uniex  10851
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