Theorem pwexd 4167

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 4166	. 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 2141   _Vcvv 2730   ~Pcpw 3566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-pw 3568

This theorem is referenced by:  fival  6947  issubm  12695  toponsspwpwg  12814  tgvalex  12844  cnpfval  12989  blfvalps  13179