Theorem pweq 3562

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 3166	. . 3 |- ( A = B -> ( x C_ A <-> x C_ B ) )
2	1	abbidv 2284	. 2 |- ( A = B -> { x | x C_ A } = { x | x C_ B } )
3		df-pw 3561	. 2 |- ~P A = { x | x C_ A }
4		df-pw 3561	. 2 |- ~P B = { x | x C_ B }
5	2, 3, 4	3eqtr4g 2224	1 |- ( A = B -> ~P A = ~P B )

Syntax hints:   -> wi 4   = wceq 1343   {cab 2151   C_ wss 3116   ~Pcpw 3559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-in 3122  df-ss 3129  df-pw 3561

This theorem is referenced by:  pweqi  3563  pweqd  3564  axpweq  4150  pwexg  4159  pwssunim  4262  ordpwsucexmid  4547  fival  6935  istopg  12637  istopon  12651  eltg  12692  tgdom  12712  ntrval  12750