Theorem nfpr 3758

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 3727	. 2 ⊢ {𝐴, 𝐵} = {𝑦 ∣ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵)}
2		nfpr.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfeq2 2404	. . . 4 ⊢ Ⅎ𝑥 𝑦 = 𝐴
4		nfpr.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
5	4	nfeq2 2404	. . . 4 ⊢ Ⅎ𝑥 𝑦 = 𝐵
6	3, 5	nfor 1627	. . 3 ⊢ Ⅎ𝑥(𝑦 = 𝐴 ∨ 𝑦 = 𝐵)
7	6	nfab 2397	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵)}
8	1, 7	nfcxfr 2389	1 ⊢ Ⅎ𝑥{𝐴, 𝐵}

Syntax hints:   ∨ wo 720   = wceq 1402  {cab 2224  Ⅎwnfc 2379  {cpr 3709

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3714  df-pr 3715

This theorem is referenced by:  nfsn  3768  nfop  3918