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Theorem axpow3 3972
Description: A variant of the Axiom of Power Sets ax-pow 3969. For any set 𝑥, there exists a set 𝑦 whose members are exactly the subsets of 𝑥 i.e. the power set of 𝑥. Axiom Pow of [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
Assertion
Ref Expression
axpow3 𝑦𝑧(𝑧𝑥𝑧𝑦)
Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem axpow3
StepHypRef Expression
1 axpow2 3971 . . 3 𝑦𝑧(𝑧𝑥𝑧𝑦)
21bm1.3ii 3920 . 2 𝑦𝑧(𝑧𝑦𝑧𝑥)
3 bicom 138 . . . 4 ((𝑧𝑥𝑧𝑦) ↔ (𝑧𝑦𝑧𝑥))
43albii 1400 . . 3 (∀𝑧(𝑧𝑥𝑧𝑦) ↔ ∀𝑧(𝑧𝑦𝑧𝑥))
54exbii 1537 . 2 (∃𝑦𝑧(𝑧𝑥𝑧𝑦) ↔ ∃𝑦𝑧(𝑧𝑦𝑧𝑥))
62, 5mpbir 144 1 𝑦𝑧(𝑧𝑥𝑧𝑦)
Colors of variables: wff set class
Syntax hints:  wb 103  wal 1283  wex 1422  wss 2983
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-11 1438  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-in 2989  df-ss 2996
This theorem is referenced by: (None)
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