Theorem nfpw 3600

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 3589	. 2 ⊢ 𝒫 𝐴 = {𝑦 ∣ 𝑦 ⊆ 𝐴}
2		nfcv 2329	. . . 4 ⊢ Ⅎ𝑥𝑦
3		nfpw.1	. . . 4 ⊢ Ⅎ𝑥𝐴
4	2, 3	nfss 3160	. . 3 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝐴
5	4	nfab 2334	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ 𝑦 ⊆ 𝐴}
6	1, 5	nfcxfr 2326	1 ⊢ Ⅎ𝑥𝒫 𝐴

Syntax hints:  {cab 2173  Ⅎwnfc 2316   ⊆ wss 3141  𝒫 cpw 3587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-in 3147  df-ss 3154  df-pw 3589

This theorem is referenced by: (None)