Theorem pweq 3629

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 3225	. . 3 |- ( A = B -> ( x C_ A <-> x C_ B ) )
2	1	abbidv 2325	. 2 |- ( A = B -> { x | x C_ A } = { x | x C_ B } )
3		df-pw 3628	. 2 |- ~P A = { x | x C_ A }
4		df-pw 3628	. 2 |- ~P B = { x | x C_ B }
5	2, 3, 4	3eqtr4g 2265	1 |- ( A = B -> ~P A = ~P B )

Syntax hints:   -> wi 4   = wceq 1373   {cab 2193   C_ wss 3174   ~Pcpw 3626

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-in 3180  df-ss 3187  df-pw 3628

This theorem is referenced by:  pweqi  3630  pweqd  3631  axpweq  4231  pwexg  4240  pwssunim  4349  ordpwsucexmid  4636  exmidpw2en  7035  fival  7098  isacnm  7346  istopg  14586  istopon  14600  eltg  14639  tgdom  14659  ntrval  14697  uhgreq12g  15787  uhgr0vb  15795  isupgren  15806  isumgren  15816