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Theorem axpow2 3976
Description: A variant of the Axiom of Power Sets ax-pow 3974 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 3974 . 2 𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦)
2 dfss2 2999 . . . . 5 (𝑧𝑥 ↔ ∀𝑤(𝑤𝑧𝑤𝑥))
32imbi1i 236 . . . 4 ((𝑧𝑥𝑧𝑦) ↔ (∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦))
43albii 1400 . . 3 (∀𝑧(𝑧𝑥𝑧𝑦) ↔ ∀𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦))
54exbii 1537 . 2 (∃𝑦𝑧(𝑧𝑥𝑧𝑦) ↔ ∃𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦))
61, 5mpbir 144 1 𝑦𝑧(𝑧𝑥𝑧𝑦)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1283  wex 1422  wss 2984
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-pow 3974
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-in 2990  df-ss 2997
This theorem is referenced by:  axpow3  3977  vpwex  3979
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