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Theorem axpow2 4766
Description: A variant of the Axiom of Power Sets ax-pow 4764 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 4764 . 2 𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦)
2 dfss2 3556 . . . . 5 (𝑧𝑥 ↔ ∀𝑤(𝑤𝑧𝑤𝑥))
32imbi1i 337 . . . 4 ((𝑧𝑥𝑧𝑦) ↔ (∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦))
43albii 1736 . . 3 (∀𝑧(𝑧𝑥𝑧𝑦) ↔ ∀𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦))
54exbii 1763 . 2 (∃𝑦𝑧(𝑧𝑥𝑧𝑦) ↔ ∃𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦))
61, 5mpbir 219 1 𝑦𝑧(𝑧𝑥𝑧𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1472  wex 1694  wss 3539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-pow 4764
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 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-in 3546  df-ss 3553
This theorem is referenced by:  axpow3  4767  pwex  4769
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