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Theorem bj-axun2 13102
Description: axun2 4352 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 13008 . . . 4 BOUNDED 𝑧𝑤
21ax-bdex 13006 . . 3 BOUNDED𝑤𝑥 𝑧𝑤
3 df-rex 2420 . . . 4 (∃𝑤𝑥 𝑧𝑤 ↔ ∃𝑤(𝑤𝑥𝑧𝑤))
4 exancom 1587 . . . 4 (∃𝑤(𝑤𝑥𝑧𝑤) ↔ ∃𝑤(𝑧𝑤𝑤𝑥))
53, 4bitri 183 . . 3 (∃𝑤𝑥 𝑧𝑤 ↔ ∃𝑤(𝑧𝑤𝑤𝑥))
62, 5bd0 13011 . 2 BOUNDED𝑤(𝑧𝑤𝑤𝑥)
7 ax-un 4350 . 2 𝑦𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦)
86, 7bdbm1.3ii 13078 1 𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑧𝑤𝑤𝑥))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wal 1329  wex 1468  wrex 2415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-un 4350  ax-bd0 13000  ax-bdex 13006  ax-bdel 13008  ax-bdsep 13071
This theorem depends on definitions:  df-bi 116  df-rex 2420
This theorem is referenced by: (None)
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