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Theorem bj-axun2 15105
Description: axun2 4450 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 15011 . . . 4 BOUNDED 𝑧𝑤
21ax-bdex 15009 . . 3 BOUNDED𝑤𝑥 𝑧𝑤
3 df-rex 2474 . . . 4 (∃𝑤𝑥 𝑧𝑤 ↔ ∃𝑤(𝑤𝑥𝑧𝑤))
4 exancom 1619 . . . 4 (∃𝑤(𝑤𝑥𝑧𝑤) ↔ ∃𝑤(𝑧𝑤𝑤𝑥))
53, 4bitri 184 . . 3 (∃𝑤𝑥 𝑧𝑤 ↔ ∃𝑤(𝑧𝑤𝑤𝑥))
62, 5bd0 15014 . 2 BOUNDED𝑤(𝑧𝑤𝑤𝑥)
7 ax-un 4448 . 2 𝑦𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦)
86, 7bdbm1.3ii 15081 1 𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑧𝑤𝑤𝑥))
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105  wal 1362  wex 1503  wrex 2469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-14 2163  ax-un 4448  ax-bd0 15003  ax-bdex 15009  ax-bdel 15011  ax-bdsep 15074
This theorem depends on definitions:  df-bi 117  df-rex 2474
This theorem is referenced by: (None)
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