Theorem pweq 3580

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 3181	. . 3 |- ( A = B -> ( x C_ A <-> x C_ B ) )
2	1	abbidv 2295	. 2 |- ( A = B -> { x | x C_ A } = { x | x C_ B } )
3		df-pw 3579	. 2 |- ~P A = { x | x C_ A }
4		df-pw 3579	. 2 |- ~P B = { x | x C_ B }
5	2, 3, 4	3eqtr4g 2235	1 |- ( A = B -> ~P A = ~P B )

Syntax hints:   -> wi 4   = wceq 1353   {cab 2163   C_ wss 3131   ~Pcpw 3577

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3137  df-ss 3144  df-pw 3579

This theorem is referenced by:  pweqi  3581  pweqd  3582  axpweq  4173  pwexg  4182  pwssunim  4286  ordpwsucexmid  4571  fival  6971  istopg  13538  istopon  13552  eltg  13591  tgdom  13611  ntrval  13649