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Theorem axpow3 4762
Description: A variant of the Axiom of Power Sets ax-pow 4759. 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 4761 . . 3 𝑦𝑧(𝑧𝑥𝑧𝑦)
21bm1.3ii 4701 . 2 𝑦𝑧(𝑧𝑦𝑧𝑥)
3 bicom 210 . . . 4 ((𝑧𝑥𝑧𝑦) ↔ (𝑧𝑦𝑧𝑥))
43albii 1735 . . 3 (∀𝑧(𝑧𝑥𝑧𝑦) ↔ ∀𝑧(𝑧𝑦𝑧𝑥))
54exbii 1762 . 2 (∃𝑦𝑧(𝑧𝑥𝑧𝑦) ↔ ∃𝑦𝑧(𝑧𝑦𝑧𝑥))
62, 5mpbir 219 1 𝑦𝑧(𝑧𝑥𝑧𝑦)
Colors of variables: wff setvar class
Syntax hints:  wb 194  wal 1472  wex 1694  wss 3534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-sep 4698  ax-pow 4759
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2591  df-cleq 2597  df-clel 2600  df-in 3541  df-ss 3548
This theorem is referenced by: (None)
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