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Theorem nfpw 3862
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 3852 . 2  |-  ~P A  =  { y  |  y 
C_  A }
2 nfcv 2571 . . . 4  |-  F/_ x
y
3 nfpw.1 . . . 4  |-  F/_ x A
42, 3nfss 3339 . . 3  |-  F/ x  y  C_  A
54nfab 2575 . 2  |-  F/_ x { y  |  y 
C_  A }
61, 5nfcxfr 2568 1  |-  F/_ x ~P A
Colors of variables: wff setvar class
Syntax hints:   {cab 2421   F/_wnfc 2558    C_ wss 3318   ~Pcpw 3850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1671  ax-6 1709  ax-7 1729  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2416
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1592  df-nf 1595  df-sb 1702  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2712  df-in 3325  df-ss 3332  df-pw 3852
This theorem is referenced by:  stoweidlem57  29700
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