Theorem pweq 3726

Description: Equality theorem for power class. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
pweq	|- (A = B -> ~PA = ~PB)

Proof of Theorem pweq

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseq2 3294	. . 3 |- (A = B -> (x C_ A <-> x C_ B))
2	1	abbidv 2468	. 2 |- (A = B -> {x | x C_ A} = {x | x C_ B})
3		df-pw 3725	. 2 |- ~PA = {x | x C_ A}
4		df-pw 3725	. 2 |- ~PB = {x | x C_ B}
5	2, 3, 4	3eqtr4g 2410	1 |- (A = B -> ~PA = ~PB)

Syntax hints:   -> wi 4   = wceq 1642  {cab 2339   C_ wss 3258  ~Pcpw 3723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-pw 3725

This theorem is referenced by:  pweqi  3727  pweqd  3728  pw1eq  4144  nnpweq  4524  sfin01  4529  sfindbl  4531  1cvsfin  4543  vfinspsslem1  4551  enpw  6088