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Theorem axpow2 4771
Description: A variant of the Axiom of Power Sets ax-pow 4769 using subset notation. Problem in [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
Assertion
Ref Expression
axpow2 𝑦𝑧(𝑧𝑥𝑧𝑦)
Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem axpow2
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 ax-pow 4769 . 2 𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦)
2 dfss2 3557 . . . . 5 (𝑧𝑥 ↔ ∀𝑤(𝑤𝑧𝑤𝑥))
32imbi1i 338 . . . 4 ((𝑧𝑥𝑧𝑦) ↔ (∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦))
43albii 1737 . . 3 (∀𝑧(𝑧𝑥𝑧𝑦) ↔ ∀𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦))
54exbii 1764 . 2 (∃𝑦𝑧(𝑧𝑥𝑧𝑦) ↔ ∃𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦))
61, 5mpbir 220 1 𝑦𝑧(𝑧𝑥𝑧𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  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-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  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:  axpow3  4772  pwex  4774
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