Theorem nfpw 3614

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 3603	. 2 ⊢ 𝒫 𝐴 = {𝑦 ∣ 𝑦 ⊆ 𝐴}
2		nfcv 2336	. . . 4 ⊢ Ⅎ𝑥𝑦
3		nfpw.1	. . . 4 ⊢ Ⅎ𝑥𝐴
4	2, 3	nfss 3172	. . 3 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝐴
5	4	nfab 2341	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ 𝑦 ⊆ 𝐴}
6	1, 5	nfcxfr 2333	1 ⊢ Ⅎ𝑥𝒫 𝐴

Syntax hints:  {cab 2179  Ⅎwnfc 2323   ⊆ wss 3153  𝒫 cpw 3601

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-in 3159  df-ss 3166  df-pw 3603

This theorem is referenced by: (None)