Theorem nfpw 3669

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 3658	. 2 |- ~P A = { y | y C_ A }
2		nfcv 2375	. . . 4 |- F/_ x y
3		nfpw.1	. . . 4 |- F/_ x A
4	2, 3	nfss 3221	. . 3 |- F/ x y C_ A
5	4	nfab 2380	. 2 |- F/_ x { y | y C_ A }
6	1, 5	nfcxfr 2372	1 |- F/_ x ~P A

Syntax hints:   {cab 2217   F/_wnfc 2362   C_ wss 3201   ~Pcpw 3656

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-in 3207  df-ss 3214  df-pw 3658

This theorem is referenced by: (None)