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Theorem bj-axun2 10973
Description: axun2 4218 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-axun2 𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑧𝑤𝑤𝑥))
Distinct variable group:   𝑥,𝑤,𝑦,𝑧

Proof of Theorem bj-axun2
StepHypRef Expression
1 ax-bdel 10879 . . . 4 BOUNDED 𝑧𝑤
21ax-bdex 10877 . . 3 BOUNDED𝑤𝑥 𝑧𝑤
3 df-rex 2359 . . . 4 (∃𝑤𝑥 𝑧𝑤 ↔ ∃𝑤(𝑤𝑥𝑧𝑤))
4 exancom 1540 . . . 4 (∃𝑤(𝑤𝑥𝑧𝑤) ↔ ∃𝑤(𝑧𝑤𝑤𝑥))
53, 4bitri 182 . . 3 (∃𝑤𝑥 𝑧𝑤 ↔ ∃𝑤(𝑧𝑤𝑤𝑥))
62, 5bd0 10882 . 2 BOUNDED𝑤(𝑧𝑤𝑤𝑥)
7 ax-un 4216 . 2 𝑦𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦)
86, 7bdbm1.3ii 10949 1 𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑧𝑤𝑤𝑥))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103  wal 1283  wex 1422  wrex 2354
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-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-un 4216  ax-bd0 10871  ax-bdex 10877  ax-bdel 10879  ax-bdsep 10942
This theorem depends on definitions:  df-bi 115  df-rex 2359
This theorem is referenced by: (None)
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