Theorem pweq 3428

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 3046	. . 3 |- ( A = B -> ( x C_ A <-> x C_ B ) )
2	1	abbidv 2205	. 2 |- ( A = B -> { x | x C_ A } = { x | x C_ B } )
3		df-pw 3427	. 2 |- ~P A = { x | x C_ A }
4		df-pw 3427	. 2 |- ~P B = { x | x C_ B }
5	2, 3, 4	3eqtr4g 2145	1 |- ( A = B -> ~P A = ~P B )

Syntax hints:   -> wi 4   = wceq 1289   {cab 2074   C_ wss 2997   ~Pcpw 3425

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 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-in 3003  df-ss 3010  df-pw 3427

This theorem is referenced by:  pweqi  3429  pweqd  3430  axpweq  3998  pwexg  4007  pwssunim  4102  ordpwsucexmid  4376