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Theorem nfpw 3939
Description: Bound-variable hypothesis builder for power class. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
Hypothesis
Ref Expression
nfpw.1  |-  F/_ x A
Assertion
Ref Expression
nfpw  |-  F/_ x ~P A

Proof of Theorem nfpw
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-pw 3929 . 2  |-  ~P A  =  { y  |  y 
C_  A }
2 nfcv 2544 . . . 4  |-  F/_ x
y
3 nfpw.1 . . . 4  |-  F/_ x A
42, 3nfss 3410 . . 3  |-  F/ x  y  C_  A
54nfab 2548 . 2  |-  F/_ x { y  |  y 
C_  A }
61, 5nfcxfr 2542 1  |-  F/_ x ~P A
Colors of variables: wff setvar class
Syntax hints:   {cab 2367   F/_wnfc 2530    C_ wss 3389   ~Pcpw 3927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ral 2737  df-in 3396  df-ss 3403  df-pw 3929
This theorem is referenced by:  esum2d  28241  stoweidlem57  32005
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