Theorem pweqi 3656

Description: Equality inference for power class. (Contributed by NM, 27-Nov-2013.)

Hypothesis

Ref	Expression
pweqi.1	|- A = B

Assertion

Ref	Expression
pweqi	|- ~P A = ~P B

Proof of Theorem pweqi

Step	Hyp	Ref	Expression
1		pweqi.1	. 2 |- A = B
2		pweq 3655	. 2 |- ( A = B -> ~P A = ~P B )
3	1, 2	ax-mp 5	1 |- ~P A = ~P B

Syntax hints:   = wceq 1397   ~Pcpw 3652

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-in 3206  df-ss 3213  df-pw 3654

This theorem is referenced by:  exmidpw  7099  exmidpweq  7100  pw1dom2  7444  pw1ne1  7446  fmelpw1o  7464  mnfnre  8221  umgrpredgv  15997  issubgr  16107  uhgrissubgr  16111