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

Theorem pweqi 3642
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 3641 . 2  |-  ( A  =  B  ->  ~P A  =  ~P B
)
31, 2ax-mp 8 1  |-  ~P A  =  ~P B
Colors of variables: wff set class
Syntax hints:    = wceq 1632   ~Pcpw 3638
This theorem is referenced by:  pwfi  7167  rankxplim  7565  pwcda1  7836  fin23lem17  7980  mnfnre  8891  qtopres  17405  hmphdis  17503  shsspwh  21841  rankeq1o  24873  onsucsuccmpi  24954  selsubf  26093  selsubf3  26094  elrfi  26872  islmodfg  27270
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-in 3172  df-ss 3179  df-pw 3640
  Copyright terms: Public domain W3C validator