Theorem nfpw 3394

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 3384	. 2 |- ~P A = { y | y C_ A }
2		nfcv 2219	. . . 4 |- F/_ x y
3		nfpw.1	. . . 4 |- F/_ x A
4	2, 3	nfss 2992	. . 3 |- F/ x y C_ A
5	4	nfab 2223	. 2 |- F/_ x { y | y C_ A }
6	1, 5	nfcxfr 2216	1 |- F/_ x ~P A

Syntax hints:   {cab 2067   F/_wnfc 2206   C_ wss 2973   ~Pcpw 3382

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-in 2979  df-ss 2986  df-pw 3384

This theorem is referenced by: (None)