Theorem nfpw 3734

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/_xA

Assertion

Ref	Expression
nfpw	|- F/_x~PA

Proof of Theorem nfpw

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-pw 3725	. 2 |- ~PA = {y | y C_ A}
2		nfcv 2490	. . . 4 |- F/_xy
3		nfpw.1	. . . 4 |- F/_xA
4	2, 3	nfss 3267	. . 3 |- F/x y C_ A
5	4	nfab 2494	. 2 |- F/_x{y | y C_ A}
6	1, 5	nfcxfr 2487	1 |- F/_x~PA

Syntax hints:  {cab 2339   F/_wnfc 2477   C_ wss 3258  ~Pcpw 3723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-pw 3725

This theorem is referenced by: (None)