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Theorem nfpw 4022
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 4012 . 2  |-  ~P A  =  { y  |  y 
C_  A }
2 nfcv 2629 . . . 4  |-  F/_ x
y
3 nfpw.1 . . . 4  |-  F/_ x A
42, 3nfss 3497 . . 3  |-  F/ x  y  C_  A
54nfab 2633 . 2  |-  F/_ x { y  |  y 
C_  A }
61, 5nfcxfr 2627 1  |-  F/_ x ~P A
Colors of variables: wff setvar class
Syntax hints:   {cab 2452   F/_wnfc 2615    C_ wss 3476   ~Pcpw 4010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-in 3483  df-ss 3490  df-pw 4012
This theorem is referenced by:  stoweidlem57  31385
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