Theorem pweq 3385

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 3021	. . 3 |- ( A = B -> ( x C_ A <-> x C_ B ) )
2	1	abbidv 2196	. 2 |- ( A = B -> { x | x C_ A } = { x | x C_ B } )
3		df-pw 3384	. 2 |- ~P A = { x | x C_ A }
4		df-pw 3384	. 2 |- ~P B = { x | x C_ B }
5	2, 3, 4	3eqtr4g 2138	1 |- ( A = B -> ~P A = ~P B )

Syntax hints:   -> wi 4   = wceq 1284   {cab 2067   C_ wss 2973   ~Pcpw 3382

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-in 2979  df-ss 2986  df-pw 3384

This theorem is referenced by:  pweqi  3386  pweqd  3387  axpweq  3945  pwex  3953  pwexg  3954  pwssunim  4039  ordpwsucexmid  4313