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Theorem axpow3 4876
Description: A variant of the Axiom of Power Sets ax-pow 4873. 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 4875 . . 3 𝑦𝑧(𝑧𝑥𝑧𝑦)
21bm1.3ii 4817 . 2 𝑦𝑧(𝑧𝑦𝑧𝑥)
3 bicom 212 . . . 4 ((𝑧𝑥𝑧𝑦) ↔ (𝑧𝑦𝑧𝑥))
43albii 1787 . . 3 (∀𝑧(𝑧𝑥𝑧𝑦) ↔ ∀𝑧(𝑧𝑦𝑧𝑥))
54exbii 1814 . 2 (∃𝑦𝑧(𝑧𝑥𝑧𝑦) ↔ ∃𝑦𝑧(𝑧𝑦𝑧𝑥))
62, 5mpbir 221 1 𝑦𝑧(𝑧𝑥𝑧𝑦)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wal 1521  wex 1744  wss 3607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-pow 4873
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-in 3614  df-ss 3621
This theorem is referenced by: (None)
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