Theorem pweq 3513

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 3121	. . 3 |- ( A = B -> ( x C_ A <-> x C_ B ) )
2	1	abbidv 2257	. 2 |- ( A = B -> { x | x C_ A } = { x | x C_ B } )
3		df-pw 3512	. 2 |- ~P A = { x | x C_ A }
4		df-pw 3512	. 2 |- ~P B = { x | x C_ B }
5	2, 3, 4	3eqtr4g 2197	1 |- ( A = B -> ~P A = ~P B )

Syntax hints:   -> wi 4   = wceq 1331   {cab 2125   C_ wss 3071   ~Pcpw 3510

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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-in 3077  df-ss 3084  df-pw 3512

This theorem is referenced by:  pweqi  3514  pweqd  3515  axpweq  4095  pwexg  4104  pwssunim  4206  ordpwsucexmid  4485  fival  6858  istopg  12166  istopon  12180  eltg  12221  tgdom  12241  ntrval  12279