Theorem nfso 4320

Description: Bound-variable hypothesis builder for total orders. (Contributed by Stefan O'Rear, 20-Jan-2015.)

Hypotheses

Ref	Expression
nfpo.r	|- F/_ x R
nfpo.a	|- F/_ x A

Assertion

Ref	Expression
nfso	|- F/ x R Or A

Proof of Theorem nfso

Dummy variables a b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iso 4315	. 2 |- ( R Or A <-> ( R Po A /\ A. a e. A A. b e. A A. c e. A ( a R b -> ( a R c \/ c R b ) ) ) )
2		nfpo.r	. . . 4 |- F/_ x R
3		nfpo.a	. . . 4 |- F/_ x A
4	2, 3	nfpo 4319	. . 3 |- F/ x R Po A
5		nfcv 2332	. . . . . . . 8 |- F/_ x a
6		nfcv 2332	. . . . . . . 8 |- F/_ x b
7	5, 2, 6	nfbr 4064	. . . . . . 7 |- F/ x a R b
8		nfcv 2332	. . . . . . . . 9 |- F/_ x c
9	5, 2, 8	nfbr 4064	. . . . . . . 8 |- F/ x a R c
10	8, 2, 6	nfbr 4064	. . . . . . . 8 |- F/ x c R b
11	9, 10	nfor 1585	. . . . . . 7 |- F/ x ( a R c \/ c R b )
12	7, 11	nfim 1583	. . . . . 6 |- F/ x ( a R b -> ( a R c \/ c R b ) )
13	3, 12	nfralxy 2528	. . . . 5 |- F/ x A. c e. A ( a R b -> ( a R c \/ c R b ) )
14	3, 13	nfralxy 2528	. . . 4 |- F/ x A. b e. A A. c e. A ( a R b -> ( a R c \/ c R b ) )
15	3, 14	nfralxy 2528	. . 3 |- F/ x A. a e. A A. b e. A A. c e. A ( a R b -> ( a R c \/ c R b ) )
16	4, 15	nfan 1576	. 2 |- F/ x ( R Po A /\ A. a e. A A. b e. A A. c e. A ( a R b -> ( a R c \/ c R b ) ) )
17	1, 16	nfxfr 1485	1 |- F/ x R Or A

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 709   F/wnf 1471   F/_wnfc 2319   A.wral 2468   class class class wbr 4018   Po wpo 4312   Or wor 4313

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-in1 615  ax-in2 616  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 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-po 4314  df-iso 4315

This theorem is referenced by: (None)