Theorem pweq 3652

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 3248	. . 3 |- ( A = B -> ( x C_ A <-> x C_ B ) )
2	1	abbidv 2347	. 2 |- ( A = B -> { x | x C_ A } = { x | x C_ B } )
3		df-pw 3651	. 2 |- ~P A = { x | x C_ A }
4		df-pw 3651	. 2 |- ~P B = { x | x C_ B }
5	2, 3, 4	3eqtr4g 2287	1 |- ( A = B -> ~P A = ~P B )

Syntax hints:   -> wi 4   = wceq 1395   {cab 2215   C_ wss 3197   ~Pcpw 3649

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-in 3203  df-ss 3210  df-pw 3651

This theorem is referenced by:  pweqi  3653  pweqd  3654  axpweq  4255  pwexg  4264  pwssunim  4375  ordpwsucexmid  4662  exmidpw2en  7074  fival  7137  isacnm  7385  istopg  14673  istopon  14687  eltg  14726  tgdom  14746  ntrval  14784  uhgreq12g  15876  uhgr0vb  15884  isupgren  15895  isumgren  15905  isuspgren  15955  isusgren  15956  isausgren  15965