Theorem nfpr 4693

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 4646	. 2 ⊢ {𝐴, 𝐵} = {𝑦 ∣ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵)}
2		nfpr.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfeq2 2920	. . . 4 ⊢ Ⅎ𝑥 𝑦 = 𝐴
4		nfpr.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
5	4	nfeq2 2920	. . . 4 ⊢ Ⅎ𝑥 𝑦 = 𝐵
6	3, 5	nfor 1907	. . 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 845   = wceq 1541  {cab 2709  Ⅎwnfc 2883  {cpr 4629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-v 3476  df-un 3952  df-sn 4628  df-pr 4630

This theorem is referenced by:  nfsn  4710  nfop  4888  nfaltop  34940