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Theorem axpow2 3929
 Description: A variant of the Axiom of Power Sets ax-pow 3927 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 3927 . 2 𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦)
2 dfss2 2934 . . . . 5 (𝑧𝑥 ↔ ∀𝑤(𝑤𝑧𝑤𝑥))
32imbi1i 227 . . . 4 ((𝑧𝑥𝑧𝑦) ↔ (∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦))
43albii 1359 . . 3 (∀𝑧(𝑧𝑥𝑧𝑦) ↔ ∀𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦))
54exbii 1496 . 2 (∃𝑦𝑧(𝑧𝑥𝑧𝑦) ↔ ∃𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦))
61, 5mpbir 134 1 𝑦𝑧(𝑧𝑥𝑧𝑦)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1241  ∃wex 1381   ⊆ wss 2917 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-pow 3927 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931 This theorem is referenced by:  axpow3  3930  pwex  3932
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