Theorem nfpw 4586

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	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfpw	⊢ Ⅎ𝑥𝒫 𝐴

Proof of Theorem nfpw

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-pw 4569	. 2 ⊢ 𝒫 𝐴 = {𝑦 ∣ 𝑦 ⊆ 𝐴}
2		nfcv 2931	. . . 4 ⊢ Ⅎ𝑥𝑦
3		nfpw.1	. . . 4 ⊢ Ⅎ𝑥𝐴
4	2, 3	nfss 3938	. . 3 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝐴
5	4	nfab 2937	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ 𝑦 ⊆ 𝐴}
6	1, 5	nfcxfr 2929	1 ⊢ Ⅎ𝑥𝒫 𝐴

Syntax hints:  {cab 2747  Ⅎwnfc 2916   ⊆ wss 3913  𝒫 cpw 4567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-ss 3930  df-pw 4569

This theorem is referenced by:  esum2d  34428  ldsysgenld  34495  stoweidlem57  46697  sge0iunmptlemre  47055  nfafv2  47878