Theorem pweq 3518

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 3126	. . 3 |- ( A = B -> ( x C_ A <-> x C_ B ) )
2	1	abbidv 2258	. 2 |- ( A = B -> { x | x C_ A } = { x | x C_ B } )
3		df-pw 3517	. 2 |- ~P A = { x | x C_ A }
4		df-pw 3517	. 2 |- ~P B = { x | x C_ B }
5	2, 3, 4	3eqtr4g 2198	1 |- ( A = B -> ~P A = ~P B )

Syntax hints:   -> wi 4   = wceq 1332   {cab 2126   C_ wss 3076   ~Pcpw 3515

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-in 3082  df-ss 3089  df-pw 3517

This theorem is referenced by:  pweqi  3519  pweqd  3520  axpweq  4103  pwexg  4112  pwssunim  4214  ordpwsucexmid  4493  fival  6866  istopg  12205  istopon  12219  eltg  12260  tgdom  12280  ntrval  12318