Theorem pweq 3659

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 3252	. . 3 |- ( A = B -> ( x C_ A <-> x C_ B ) )
2	1	abbidv 2350	. 2 |- ( A = B -> { x | x C_ A } = { x | x C_ B } )
3		df-pw 3658	. 2 |- ~P A = { x | x C_ A }
4		df-pw 3658	. 2 |- ~P B = { x | x C_ B }
5	2, 3, 4	3eqtr4g 2289	1 |- ( A = B -> ~P A = ~P B )

Syntax hints:   -> wi 4   = wceq 1398   {cab 2217   C_ wss 3201   ~Pcpw 3656

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 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-in 3207  df-ss 3214  df-pw 3658

This theorem is referenced by:  pweqi  3660  pweqd  3661  axpweq  4267  pwexg  4276  pwssunim  4387  ordpwsucexmid  4674  exmidpw2en  7147  fival  7229  isacnm  7478  istopg  14810  istopon  14824  eltg  14863  tgdom  14883  ntrval  14921  uhgreq12g  16017  uhgr0vb  16025  isupgren  16036  isumgren  16046  isuspgren  16098  isusgren  16099  isausgren  16108