Theorem pweq 3436

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 3049	. . 3 |- ( A = B -> ( x C_ A <-> x C_ B ) )
2	1	abbidv 2206	. 2 |- ( A = B -> { x | x C_ A } = { x | x C_ B } )
3		df-pw 3435	. 2 |- ~P A = { x | x C_ A }
4		df-pw 3435	. 2 |- ~P B = { x | x C_ B }
5	2, 3, 4	3eqtr4g 2146	1 |- ( A = B -> ~P A = ~P B )

Syntax hints:   -> wi 4   = wceq 1290   {cab 2075   C_ wss 3000   ~Pcpw 3433

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-11 1443  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-in 3006  df-ss 3013  df-pw 3435

This theorem is referenced by:  pweqi  3437  pweqd  3438  axpweq  4012  pwexg  4021  pwssunim  4120  ordpwsucexmid  4399  istopg  11759  istopon  11773  eltg  11813  tgdom  11833  ntrval  11871