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Theorem axpow3 3921
Description: A variant of the Axiom of Power Sets ax-pow 3918. For any set x, there exists a set y whose members are exactly the subsets of x i.e. the power set of x. Axiom Pow of [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
Assertion
Ref Expression
axpow3 yz(zxz y)
Distinct variable group:   x,y,z

Proof of Theorem axpow3
StepHypRef Expression
1 axpow2 3920 . . 3 yz(zxz y)
21bm1.3ii 3869 . 2 yz(z yzx)
3 bicom 128 . . . 4 ((zxz y) ↔ (z yzx))
43albii 1356 . . 3 (z(zxz y) ↔ z(z yzx))
54exbii 1493 . 2 (yz(zxz y) ↔ yz(z yzx))
62, 5mpbir 134 1 yz(zxz y)
Colors of variables: wff set class
Syntax hints:  wb 98  wal 1240  wex 1378  wss 2911
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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-in 2918  df-ss 2925
This theorem is referenced by: (None)
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