Theorem pweqd 3610

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 3608	. 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 3605

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-in 3163  df-ss 3170  df-pw 3607

This theorem is referenced by:  pmvalg  6718  issubm  13104  issubg  13303  subgex  13306  issubrng  13755  issubrg  13777  lsssetm  13912  lspfval  13944  lsppropd  13988  sraval  13993  basis1  14283  baspartn  14286  cldval  14335  ntrfval  14336  clsfval  14337  neifval  14376  mopnfss  14683