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Theorem pwexg 3907
Description: Power set axiom expressed in class notation, with the sethood requirement as an antecedent. Axiom 4 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Oct-2003.)
Assertion
Ref Expression
pwexg (A 𝑉 → 𝒫 A V)

Proof of Theorem pwexg
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 pweq 3337 . . 3 (x = A → 𝒫 x = 𝒫 A)
21eleq1d 2088 . 2 (x = A → (𝒫 x V ↔ 𝒫 A V))
3 vex 2538 . . 3 x V
43pwex 3906 . 2 𝒫 x V
52, 4vtoclg 2590 1 (A 𝑉 → 𝒫 A V)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1228   wcel 1374  Vcvv 2535  𝒫 cpw 3334
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-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-in 2901  df-ss 2908  df-pw 3336
This theorem is referenced by:  abssexg  3908  snexgOLD  3909  snexg  3910  pwel  3928  uniexb  4155  xpexg  4379  fabexg  5002
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