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Theorem axpow2 3903
Description: A variant of the Axiom of Power Sets ax-pow 3901 using subset notation. Problem in {BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
Assertion
Ref Expression
axpow2 yz(zxz y)
Distinct variable group:   x,y,z

Proof of Theorem axpow2
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 ax-pow 3901 . 2 yz(w(w zw x) → z y)
2 dfss2 2911 . . . . 5 (zxw(w zw x))
32imbi1i 227 . . . 4 ((zxz y) ↔ (w(w zw x) → z y))
43albii 1339 . . 3 (z(zxz y) ↔ z(w(w zw x) → z y))
54exbii 1478 . 2 (yz(zxz y) ↔ yz(w(w zw x) → z y))
61, 5mpbir 134 1 yz(zxz y)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1226  wex 1362  wss 2894
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-11 1378  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-pow 3901
This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-in 2901  df-ss 2908
This theorem is referenced by:  axpow3  3904  pwex  3906
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