Theorem pweq 3362

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 2967	. . 3 |- ( A = B -> ( x C_ A <-> x C_ B ) )
2	1	abbidv 2155	. 2 |- ( A = B -> { x | x C_ A } = { x | x C_ B } )
3		df-pw 3361	. 2 |- ~P A = { x | x C_ A }
4		df-pw 3361	. 2 |- ~P B = { x | x C_ B }
5	2, 3, 4	3eqtr4g 2097	1 |- ( A = B -> ~P A = ~P B )

Syntax hints:   -> wi 4   = wceq 1243   {cab 2026   C_ wss 2917   ~Pcpw 3359

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931  df-pw 3361

This theorem is referenced by:  pweqi  3363  pweqd  3364  axpweq  3924  pwex  3932  pwexg  3933  pwssunim  4021  ordpwsucexmid  4294