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Theorem nfpw 3812
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 3803 . 2  |-  ~P A  =  { y  |  y 
C_  A }
2 nfcv 2574 . . . 4  |-  F/_ x
y
3 nfpw.1 . . . 4  |-  F/_ x A
42, 3nfss 3343 . . 3  |-  F/ x  y  C_  A
54nfab 2578 . 2  |-  F/_ x { y  |  y 
C_  A }
61, 5nfcxfr 2571 1  |-  F/_ x ~P A
Colors of variables: wff set class
Syntax hints:   {cab 2424   F/_wnfc 2561    C_ wss 3322   ~Pcpw 3801
This theorem is referenced by:  stoweidlem57  27784
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-in 3329  df-ss 3336  df-pw 3803
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