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Theorem axpow3 4772
 Description: A variant of the Axiom of Power Sets ax-pow 4769. 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 4771 . . 3 𝑦𝑧(𝑧𝑥𝑧𝑦)
21bm1.3ii 4712 . 2 𝑦𝑧(𝑧𝑦𝑧𝑥)
3 bicom 211 . . . 4 ((𝑧𝑥𝑧𝑦) ↔ (𝑧𝑦𝑧𝑥))
43albii 1737 . . 3 (∀𝑧(𝑧𝑥𝑧𝑦) ↔ ∀𝑧(𝑧𝑦𝑧𝑥))
54exbii 1764 . 2 (∃𝑦𝑧(𝑧𝑥𝑧𝑦) ↔ ∃𝑦𝑧(𝑧𝑦𝑧𝑥))
62, 5mpbir 220 1 𝑦𝑧(𝑧𝑥𝑧𝑦)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195  ∀wal 1473  ∃wex 1695   ⊆ wss 3540 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-pow 4769 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-in 3547  df-ss 3554 This theorem is referenced by: (None)
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