Theorem pwexd 4313

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

Hypothesis

Ref	Expression
pwexd.1	|- ( ph -> A e. V )

Assertion

Ref	Expression
pwexd	|- ( ph -> ~P A e. _V )

Proof of Theorem pwexd

Step	Hyp	Ref	Expression
1		pwexd.1	. 2 |- ( ph -> A e. V )
2		pwexg 4312	. 2 |- ( A e. V -> ~P A e. _V )
3	1, 2	syl 14	1 |- ( ph -> ~P A e. _V )

Syntax hints:   -> wi 4   e. wcel 2209   _Vcvv 2821   ~Pcpw 3685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-pw 3687

This theorem is referenced by:  fival  7294  hashfibclem  11260  tgvalex  13594  issubm  13756  issubg  13953  subgex  13956  issubrng  14480  issubrg  14502  lssex  14663  lsssetm  14665  lspfval  14697  lspex  14704  sraval  14746  toponsspwpwg  15046  cnpfval  15219  blfvalps  15409