Theorem pweq 3569

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 3171	. . 3 |- ( A = B -> ( x C_ A <-> x C_ B ) )
2	1	abbidv 2288	. 2 |- ( A = B -> { x | x C_ A } = { x | x C_ B } )
3		df-pw 3568	. 2 |- ~P A = { x | x C_ A }
4		df-pw 3568	. 2 |- ~P B = { x | x C_ B }
5	2, 3, 4	3eqtr4g 2228	1 |- ( A = B -> ~P A = ~P B )

Syntax hints:   -> wi 4   = wceq 1348   {cab 2156   C_ wss 3121   ~Pcpw 3566

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-in 3127  df-ss 3134  df-pw 3568

This theorem is referenced by:  pweqi  3570  pweqd  3571  axpweq  4157  pwexg  4166  pwssunim  4269  ordpwsucexmid  4554  fival  6947  istopg  12791  istopon  12805  eltg  12846  tgdom  12866  ntrval  12904