Theorem nfpw 3523

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 3512	. 2 |- ~P A = { y | y C_ A }
2		nfcv 2281	. . . 4 |- F/_ x y
3		nfpw.1	. . . 4 |- F/_ x A
4	2, 3	nfss 3090	. . 3 |- F/ x y C_ A
5	4	nfab 2286	. 2 |- F/_ x { y | y C_ A }
6	1, 5	nfcxfr 2278	1 |- F/_ x ~P A

Syntax hints:   {cab 2125   F/_wnfc 2268   C_ wss 3071   ~Pcpw 3510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-in 3077  df-ss 3084  df-pw 3512

This theorem is referenced by: (None)