Theorem pweq 3619

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 3217	. . 3 |- ( A = B -> ( x C_ A <-> x C_ B ) )
2	1	abbidv 2323	. 2 |- ( A = B -> { x | x C_ A } = { x | x C_ B } )
3		df-pw 3618	. 2 |- ~P A = { x | x C_ A }
4		df-pw 3618	. 2 |- ~P B = { x | x C_ B }
5	2, 3, 4	3eqtr4g 2263	1 |- ( A = B -> ~P A = ~P B )

Syntax hints:   -> wi 4   = wceq 1373   {cab 2191   C_ wss 3166   ~Pcpw 3616

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-in 3172  df-ss 3179  df-pw 3618

This theorem is referenced by:  pweqi  3620  pweqd  3621  axpweq  4216  pwexg  4225  pwssunim  4332  ordpwsucexmid  4619  exmidpw2en  7011  fival  7074  isacnm  7317  istopg  14504  istopon  14518  eltg  14557  tgdom  14577  ntrval  14615