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Theorem bj-uniex2 11464
Description: uniex2 4254 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 11424 . . . 4 BOUNDED 𝑥
21bdeli 11394 . . 3 BOUNDED 𝑧 𝑥
3 zfun 4252 . . . 4 𝑦𝑧(∃𝑦(𝑧𝑦𝑦𝑥) → 𝑧𝑦)
4 eluni 3651 . . . . . . 7 (𝑧 𝑥 ↔ ∃𝑦(𝑧𝑦𝑦𝑥))
54imbi1i 236 . . . . . 6 ((𝑧 𝑥𝑧𝑦) ↔ (∃𝑦(𝑧𝑦𝑦𝑥) → 𝑧𝑦))
65albii 1404 . . . . 5 (∀𝑧(𝑧 𝑥𝑧𝑦) ↔ ∀𝑧(∃𝑦(𝑧𝑦𝑦𝑥) → 𝑧𝑦))
76exbii 1541 . . . 4 (∃𝑦𝑧(𝑧 𝑥𝑧𝑦) ↔ ∃𝑦𝑧(∃𝑦(𝑧𝑦𝑦𝑥) → 𝑧𝑦))
83, 7mpbir 144 . . 3 𝑦𝑧(𝑧 𝑥𝑧𝑦)
92, 8bdbm1.3ii 11439 . 2 𝑦𝑧(𝑧𝑦𝑧 𝑥)
10 dfcleq 2082 . . 3 (𝑦 = 𝑥 ↔ ∀𝑧(𝑧𝑦𝑧 𝑥))
1110exbii 1541 . 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 1287   = wceq 1289  wex 1426  wcel 1438   cuni 3648
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-un 4251  ax-bd0 11361  ax-bdex 11367  ax-bdel 11369  ax-bdsb 11370  ax-bdsep 11432
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-uni 3649  df-bdc 11389
This theorem is referenced by:  bj-uniex  11465
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