Theorem pweqd 3606

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

Hypothesis

Ref	Expression
pweqd.1	|- ( ph -> A = B )

Assertion

Ref	Expression
pweqd	|- ( ph -> ~P A = ~P B )

Proof of Theorem pweqd

Step	Hyp	Ref	Expression
1		pweqd.1	. 2 |- ( ph -> A = B )
2		pweq 3604	. 2 |- ( A = B -> ~P A = ~P B )
3	1, 2	syl 14	1 |- ( ph -> ~P A = ~P B )

Syntax hints:   -> wi 4   = wceq 1364   ~Pcpw 3601

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-in 3159  df-ss 3166  df-pw 3603

This theorem is referenced by:  pmvalg  6713  issubm  13044  issubg  13243  subgex  13246  issubrng  13695  issubrg  13717  lsssetm  13852  lspfval  13884  lsppropd  13928  sraval  13933  basis1  14215  baspartn  14218  cldval  14267  ntrfval  14268  clsfval  14269  neifval  14308  mopnfss  14615