Theorem nfpr 3723

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 3692	. 2 |- { A , B } = { y | ( y = A \/ y = B ) }
2		nfpr.1	. . . . 5 |- F/_ x A
3	2	nfeq2 2387	. . . 4 |- F/ x y = A
4		nfpr.2	. . . . 5 |- F/_ x B
5	4	nfeq2 2387	. . . 4 |- F/ x y = B
6	3, 5	nfor 1623	. . 3 |- F/ x ( y = A \/ y = B )
7	6	nfab 2380	. 2 |- F/_ x { y | ( y = A \/ y = B ) }
8	1, 7	nfcxfr 2372	1 |- F/_ x { A , B }

Syntax hints:   \/ wo 716   = wceq 1398   {cab 2217   F/_wnfc 2362   {cpr 3674

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680

This theorem is referenced by:  nfsn  3733  nfop  3883