Theorem pweqd 3607

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 3605	. 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 3602

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 3160  df-ss 3167  df-pw 3604

This theorem is referenced by:  pmvalg  6715  issubm  13047  issubg  13246  subgex  13249  issubrng  13698  issubrg  13720  lsssetm  13855  lspfval  13887  lsppropd  13931  sraval  13936  basis1  14226  baspartn  14229  cldval  14278  ntrfval  14279  clsfval  14280  neifval  14319  mopnfss  14626