Theorem nfpr 3668

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 3637	. 2 |- { A , B } = { y | ( y = A \/ y = B ) }
2		nfpr.1	. . . . 5 |- F/_ x A
3	2	nfeq2 2348	. . . 4 |- F/ x y = A
4		nfpr.2	. . . . 5 |- F/_ x B
5	4	nfeq2 2348	. . . 4 |- F/ x y = B
6	3, 5	nfor 1585	. . 3 |- F/ x ( y = A \/ y = B )
7	6	nfab 2341	. 2 |- F/_ x { y | ( y = A \/ y = B ) }
8	1, 7	nfcxfr 2333	1 |- F/_ x { A , B }

Syntax hints:   \/ wo 709   = wceq 1364   {cab 2179   F/_wnfc 2323   {cpr 3619

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625

This theorem is referenced by:  nfsn  3678  nfop  3820