Theorem pwexd 5318

Description: Deduction version of the power set axiom. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypothesis

Ref	Expression
pwexd.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)

Assertion

Ref	Expression
pwexd	⊢ (𝜑 → 𝒫 𝐴 ∈ V)

Proof of Theorem pwexd

Step	Hyp	Ref	Expression
1		pwexd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		pwexg 5317	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝒫 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2109  Vcvv 3436  𝒫 cpw 4551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5235  ax-pow 5304

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3438  df-ss 3920  df-pw 4553

This theorem is referenced by:  fabexd  7870  undefval  8209  mapexOLD  8759  pmvalg  8764  fopwdom  9002  pwdom  9046  fineqvlem  9155  fival  9302  fipwuni  9316  hartogslem2  9435  wdompwdom  9470  harwdom  9483  canthwe  10545  canthp1lem2  10547  gchdjuidm  10562  gchpwdom  10564  gchhar  10573  prdsmulr  17363  selvffval  22018  toponsspwpw  22807  mretopd  22977  ordtbaslem  23073  ptcmplem1  23937  isust  24089  blfvalps  24269  carsgval  34277  neibastop2lem  36344  bj-imdirvallem  37164  bj-imdirval2lem  37166  rfovcnvf1od  43987  fsovfd  43995  fsovcnvlem  43996  dssmapnvod  44003  dssmapf1od  44004  ntrneif1o  44058  ntrneicnv  44061  ntrneiel  44064  clsneiel1  44091  neicvgf1o  44097  neicvgnvo  44098  neicvgel1  44102  ntrelmap  44108  clselmap  44110  salexct  46325  psmeasurelem  46461  caragenval  46484  omeunile  46496  0ome  46520  isomennd  46522  afv2ex  47208  gpgvtx  48037  gpgiedg  48038