Theorem pwexd 5325

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 5324	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝒫 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2114  Vcvv 3441  𝒫 cpw 4555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5242  ax-pow 5311

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3443  df-ss 3919  df-pw 4557

This theorem is referenced by:  fabexd  7881  undefval  8220  mapexOLD  8773  pmvalg  8778  fopwdom  9017  pwdom  9061  fineqvlem  9170  fival  9319  fipwuni  9333  hartogslem2  9452  wdompwdom  9487  harwdom  9500  canthwe  10566  canthp1lem2  10568  gchdjuidm  10583  gchpwdom  10585  gchhar  10594  prdsmulr  17383  selvffval  22080  toponsspwpw  22870  mretopd  23040  ordtbaslem  23136  ptcmplem1  24000  isust  24152  blfvalps  24331  esplympl  33727  carsgval  34462  neibastop2lem  36556  bj-imdirvallem  37387  bj-imdirval2lem  37389  rfovcnvf1od  44312  fsovfd  44320  fsovcnvlem  44321  dssmapnvod  44328  dssmapf1od  44329  ntrneif1o  44383  ntrneicnv  44386  ntrneiel  44389  clsneiel1  44416  neicvgf1o  44422  neicvgnvo  44423  neicvgel1  44427  ntrelmap  44433  clselmap  44435  salexct  46645  psmeasurelem  46781  caragenval  46804  omeunile  46816  0ome  46840  isomennd  46842  afv2ex  47527  gpgvtx  48356  gpgiedg  48357