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Theorem pwexg 3924
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 3354 . . 3 (x = A → 𝒫 x = 𝒫 A)
21eleq1d 2103 . 2 (x = A → (𝒫 x V ↔ 𝒫 A V))
3 vex 2554 . . 3 x V
43pwex 3923 . 2 𝒫 x V
52, 4vtoclg 2607 1 (A 𝑉 → 𝒫 A V)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   wcel 1390  Vcvv 2551  𝒫 cpw 3351
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  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-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353
This theorem is referenced by:  abssexg  3925  snexgOLD  3926  snexg  3927  pwel  3945  uniexb  4171  xpexg  4395  fabexg  5020  fopwdom  6246
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