Theorem nfpw 3629

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.

Step	Hyp	Ref	Expression
1		df-pw 3618	. 2 |- ~P A = { y | y C_ A }
2		nfcv 2348	. . . 4 |- F/_ x y
3		nfpw.1	. . . 4 |- F/_ x A
4	2, 3	nfss 3186	. . 3 |- F/ x y C_ A
5	4	nfab 2353	. 2 |- F/_ x { y | y C_ A }
6	1, 5	nfcxfr 2345	1 |- F/_ x ~P A

Syntax hints:   {cab 2191   F/_wnfc 2335   C_ wss 3166   ~Pcpw 3616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-in 3172  df-ss 3179  df-pw 3618

This theorem is referenced by: (None)