Theorem pwexd 4265

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 4264	. 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 2200   _Vcvv 2799   ~Pcpw 3649

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203  df-ss 3210  df-pw 3651

This theorem is referenced by:  fival  7137  tgvalex  13296  issubm  13505  issubg  13710  subgex  13713  issubrng  14163  issubrg  14185  lssex  14318  lsssetm  14320  lspfval  14352  lspex  14359  sraval  14401  toponsspwpwg  14696  cnpfval  14869  blfvalps  15059