Theorem nfpw 3579

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 3568	. 2 |- ~P A = { y | y C_ A }
2		nfcv 2312	. . . 4 |- F/_ x y
3		nfpw.1	. . . 4 |- F/_ x A
4	2, 3	nfss 3140	. . 3 |- F/ x y C_ A
5	4	nfab 2317	. 2 |- F/_ x { y | y C_ A }
6	1, 5	nfcxfr 2309	1 |- F/_ x ~P A

Syntax hints:   {cab 2156   F/_wnfc 2299   C_ wss 3121   ~Pcpw 3566

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-in 3127  df-ss 3134  df-pw 3568

This theorem is referenced by: (None)