Theorem nfpr 4673

Description: Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.)

Hypotheses

Ref	Expression
nfpr.1	⊢ Ⅎ𝑥𝐴
nfpr.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
nfpr	⊢ Ⅎ𝑥{𝐴, 𝐵}

Proof of Theorem nfpr

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfpr2 4627	. 2 ⊢ {𝐴, 𝐵} = {𝑦 ∣ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵)}
2		nfpr.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfeq2 2917	. . . 4 ⊢ Ⅎ𝑥 𝑦 = 𝐴
4		nfpr.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
5	4	nfeq2 2917	. . . 4 ⊢ Ⅎ𝑥 𝑦 = 𝐵
6	3, 5	nfor 1904	. . 3 ⊢ Ⅎ𝑥(𝑦 = 𝐴 ∨ 𝑦 = 𝐵)
7	6	nfab 2905	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵)}
8	1, 7	nfcxfr 2897	1 ⊢ Ⅎ𝑥{𝐴, 𝐵}

Syntax hints:   ∨ wo 847   = wceq 1540  {cab 2714  Ⅎwnfc 2884  {cpr 4608

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-v 3466  df-un 3936  df-sn 4607  df-pr 4609

This theorem is referenced by:  nfsn  4688  nfop  4870  nfaltop  36003