Theorem nfpw 4594

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 4577	. 2 ⊢ 𝒫 𝐴 = {𝑦 ∣ 𝑦 ⊆ 𝐴}
2		nfcv 2898	. . . 4 ⊢ Ⅎ𝑥𝑦
3		nfpw.1	. . . 4 ⊢ Ⅎ𝑥𝐴
4	2, 3	nfss 3951	. . 3 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝐴
5	4	nfab 2904	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ 𝑦 ⊆ 𝐴}
6	1, 5	nfcxfr 2896	1 ⊢ Ⅎ𝑥𝒫 𝐴

Syntax hints:  {cab 2713  Ⅎwnfc 2883   ⊆ wss 3926  𝒫 cpw 4575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-ss 3943  df-pw 4577

This theorem is referenced by:  esum2d  34124  ldsysgenld  34191  stoweidlem57  46086  sge0iunmptlemre  46444  nfafv2  47247