Theorem pweq 3677

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 3266	. . 3 |- ( A = B -> ( x C_ A <-> x C_ B ) )
2	1	abbidv 2354	. 2 |- ( A = B -> { x | x C_ A } = { x | x C_ B } )
3		df-pw 3676	. 2 |- ~P A = { x | x C_ A }
4		df-pw 3676	. 2 |- ~P B = { x | x C_ B }
5	2, 3, 4	3eqtr4g 2292	1 |- ( A = B -> ~P A = ~P B )

Syntax hints:   -> wi 4   = wceq 1398   {cab 2220   C_ wss 3214   ~Pcpw 3674

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-in 3220  df-ss 3227  df-pw 3676

This theorem is referenced by:  pweqi  3678  pweqd  3679  axpweq  4289  pwexg  4298  pwssunim  4410  ordpwsucexmid  4697  exmidpw2en  7185  fival  7270  isacnm  7523  hashfibc  11232  istopg  14990  istopon  15004  eltg  15043  tgdom  15063  ntrval  15101  uhgreq12g  16197  uhgr0vb  16205  isupgren  16216  isumgren  16226  isuspgren  16278  isusgren  16279  isausgren  16288