Theorem nfpr 3683

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 3652	. 2 |- { A , B } = { y | ( y = A \/ y = B ) }
2		nfpr.1	. . . . 5 |- F/_ x A
3	2	nfeq2 2360	. . . 4 |- F/ x y = A
4		nfpr.2	. . . . 5 |- F/_ x B
5	4	nfeq2 2360	. . . 4 |- F/ x y = B
6	3, 5	nfor 1597	. . 3 |- F/ x ( y = A \/ y = B )
7	6	nfab 2353	. 2 |- F/_ x { y | ( y = A \/ y = B ) }
8	1, 7	nfcxfr 2345	1 |- F/_ x { A , B }

Syntax hints:   \/ wo 710   = wceq 1373   {cab 2191   F/_wnfc 2335   {cpr 3634

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640

This theorem is referenced by:  nfsn  3693  nfop  3835