Theorem nfpr 4697

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 4651	. 2 ⊢ {𝐴, 𝐵} = {𝑦 ∣ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵)}
2		nfpr.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfeq2 2921	. . . 4 ⊢ Ⅎ𝑥 𝑦 = 𝐴
4		nfpr.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
5	4	nfeq2 2921	. . . 4 ⊢ Ⅎ𝑥 𝑦 = 𝐵
6	3, 5	nfor 1902	. . 3 ⊢ Ⅎ𝑥(𝑦 = 𝐴 ∨ 𝑦 = 𝐵)
7	6	nfab 2909	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵)}
8	1, 7	nfcxfr 2901	1 ⊢ Ⅎ𝑥{𝐴, 𝐵}

Syntax hints:   ∨ wo 847   = wceq 1537  {cab 2712  Ⅎwnfc 2888  {cpr 4633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-v 3480  df-un 3968  df-sn 4632  df-pr 4634

This theorem is referenced by:  nfsn  4712  nfop  4894  nfaltop  35962