Theorem nfpr 3568

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 3541	. 2 |- { A , B } = { y | ( y = A \/ y = B ) }
2		nfpr.1	. . . . 5 |- F/_ x A
3	2	nfeq2 2291	. . . 4 |- F/ x y = A
4		nfpr.2	. . . . 5 |- F/_ x B
5	4	nfeq2 2291	. . . 4 |- F/ x y = B
6	3, 5	nfor 1553	. . 3 |- F/ x ( y = A \/ y = B )
7	6	nfab 2284	. 2 |- F/_ x { y | ( y = A \/ y = B ) }
8	1, 7	nfcxfr 2276	1 |- F/_ x { A , B }

Syntax hints:   \/ wo 697   = wceq 1331   {cab 2123   F/_wnfc 2266   {cpr 3523

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529

This theorem is referenced by:  nfsn  3578  nfop  3716