MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pweqi Structured version   Unicode version

Theorem pweqi 4014
Description: Equality inference for power class. (Contributed by NM, 27-Nov-2013.)
Hypothesis
Ref Expression
pweqi.1  |-  A  =  B
Assertion
Ref Expression
pweqi  |-  ~P A  =  ~P B

Proof of Theorem pweqi
StepHypRef Expression
1 pweqi.1 . 2  |-  A  =  B
2 pweq 4013 . 2  |-  ( A  =  B  ->  ~P A  =  ~P B
)
31, 2ax-mp 5 1  |-  ~P A  =  ~P B
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379   ~Pcpw 4010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-in 3483  df-ss 3490  df-pw 4012
This theorem is referenced by:  pwfi  7811  rankxplim  8293  pwcda1  8570  fin23lem17  8714  mnfnre  9632  qtopres  19931  hmphdis  20029  ust0  20454  shsspwh  25837  circtopn  27635  rankeq1o  29402  onsucsuccmpi  29482  elrfi  30228  islmodfg  30619  uhgrepe  31847  lcoop  32085  lincvalsc0  32095  linc0scn0  32097  lincdifsn  32098  linc1  32099  lspsslco  32111  lincresunit3lem2  32154  lincresunit3  32155
  Copyright terms: Public domain W3C validator