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Theorem axpow2 5259
Description: A variant of the Axiom of Power Sets ax-pow 5257 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 5257 . 2 𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦)
2 dfss2 3952 . . . . 5 (𝑧𝑥 ↔ ∀𝑤(𝑤𝑧𝑤𝑥))
32imbi1i 351 . . . 4 ((𝑧𝑥𝑧𝑦) ↔ (∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦))
43albii 1811 . . 3 (∀𝑧(𝑧𝑥𝑧𝑦) ↔ ∀𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦))
54exbii 1839 . 2 (∃𝑦𝑧(𝑧𝑥𝑧𝑦) ↔ ∃𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦))
61, 5mpbir 232 1 𝑦𝑧(𝑧𝑥𝑧𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1526  wex 1771  wss 3933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-pow 5257
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-in 3940  df-ss 3949
This theorem is referenced by:  axpow3  5260  vpwex  5269
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