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Theorem nfpw 3893
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 3883 . 2  |-  ~P A  =  { y  |  y 
C_  A }
2 nfcv 2589 . . . 4  |-  F/_ x
y
3 nfpw.1 . . . 4  |-  F/_ x A
42, 3nfss 3370 . . 3  |-  F/ x  y  C_  A
54nfab 2593 . 2  |-  F/_ x { y  |  y 
C_  A }
61, 5nfcxfr 2587 1  |-  F/_ x ~P A
Colors of variables: wff setvar class
Syntax hints:   {cab 2429   F/_wnfc 2575    C_ wss 3349   ~Pcpw 3881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ral 2741  df-in 3356  df-ss 3363  df-pw 3883
This theorem is referenced by:  stoweidlem57  29878
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