Theorem pweqd 3631

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 3629	. 2 |- ( A = B -> ~P A = ~P B )
3	1, 2	syl 14	1 |- ( ph -> ~P A = ~P B )

Syntax hints:   -> wi 4   = wceq 1373   ~Pcpw 3626

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-in 3180  df-ss 3187  df-pw 3628

This theorem is referenced by:  pmvalg  6769  issubm  13419  issubg  13624  subgex  13627  issubrng  14076  issubrg  14098  lsssetm  14233  lspfval  14265  lsppropd  14309  sraval  14314  basis1  14634  baspartn  14637  cldval  14686  ntrfval  14687  clsfval  14688  neifval  14727  mopnfss  15034  isuhgrm  15782  isushgrm  15783  isuhgropm  15792  uhgrun  15797  isupgren  15806  upgrop  15815  isumgren  15816  upgrun  15832  umgrun  15834