Theorem pweq 3672

Description: Equality theorem for power class. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
pweq	|- ( A = B -> ~P A = ~P B )

Proof of Theorem pweq

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseq2 3262	. . 3 |- ( A = B -> ( x C_ A <-> x C_ B ) )
2	1	abbidv 2352	. 2 |- ( A = B -> { x | x C_ A } = { x | x C_ B } )
3		df-pw 3671	. 2 |- ~P A = { x | x C_ A }
4		df-pw 3671	. 2 |- ~P B = { x | x C_ B }
5	2, 3, 4	3eqtr4g 2290	1 |- ( A = B -> ~P A = ~P B )

Syntax hints:   -> wi 4   = wceq 1398   {cab 2218   C_ wss 3211   ~Pcpw 3669

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-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-in 3217  df-ss 3224  df-pw 3671

This theorem is referenced by:  pweqi  3673  pweqd  3674  axpweq  4284  pwexg  4293  pwssunim  4405  ordpwsucexmid  4692  exmidpw2en  7172  fival  7257  isacnm  7510  hashfibc  11207  istopg  14864  istopon  14878  eltg  14917  tgdom  14937  ntrval  14975  uhgreq12g  16071  uhgr0vb  16079  isupgren  16090  isumgren  16100  isuspgren  16152  isusgren  16153  isausgren  16162