Theorem pweq 3604

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 3203	. . 3 |- ( A = B -> ( x C_ A <-> x C_ B ) )
2	1	abbidv 2311	. 2 |- ( A = B -> { x | x C_ A } = { x | x C_ B } )
3		df-pw 3603	. 2 |- ~P A = { x | x C_ A }
4		df-pw 3603	. 2 |- ~P B = { x | x C_ B }
5	2, 3, 4	3eqtr4g 2251	1 |- ( A = B -> ~P A = ~P B )

Syntax hints:   -> wi 4   = wceq 1364   {cab 2179   C_ wss 3153   ~Pcpw 3601

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-in 3159  df-ss 3166  df-pw 3603

This theorem is referenced by:  pweqi  3605  pweqd  3606  axpweq  4200  pwexg  4209  pwssunim  4315  ordpwsucexmid  4602  exmidpw2en  6968  fival  7029  istopg  14167  istopon  14181  eltg  14220  tgdom  14240  ntrval  14278