Theorem nfpr 3626

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

Hypotheses

Ref	Expression
nfpr.1	|- F/_ x A
nfpr.2	|- F/_ x B

Assertion

Ref	Expression
nfpr	|- F/_ x { A , B }

Proof of Theorem nfpr

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfpr2 3595	. 2 |- { A , B } = { y | ( y = A \/ y = B ) }
2		nfpr.1	. . . . 5 |- F/_ x A
3	2	nfeq2 2320	. . . 4 |- F/ x y = A
4		nfpr.2	. . . . 5 |- F/_ x B
5	4	nfeq2 2320	. . . 4 |- F/ x y = B
6	3, 5	nfor 1562	. . 3 |- F/ x ( y = A \/ y = B )
7	6	nfab 2313	. 2 |- F/_ x { y | ( y = A \/ y = B ) }
8	1, 7	nfcxfr 2305	1 |- F/_ x { A , B }

Syntax hints:   \/ wo 698   = wceq 1343   {cab 2151   F/_wnfc 2295   {cpr 3577

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583

This theorem is referenced by:  nfsn  3636  nfop  3774