Theorem nfso 4120

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 4115	. 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 4119	. . 3 |- F/ x R Po A
5		nfcv 2228	. . . . . . . 8 |- F/_ x a
6		nfcv 2228	. . . . . . . 8 |- F/_ x b
7	5, 2, 6	nfbr 3881	. . . . . . 7 |- F/ x a R b
8		nfcv 2228	. . . . . . . . 9 |- F/_ x c
9	5, 2, 8	nfbr 3881	. . . . . . . 8 |- F/ x a R c
10	8, 2, 6	nfbr 3881	. . . . . . . 8 |- F/ x c R b
11	9, 10	nfor 1511	. . . . . . 7 |- F/ x ( a R c \/ c R b )
12	7, 11	nfim 1509	. . . . . 6 |- F/ x ( a R b -> ( a R c \/ c R b ) )
13	3, 12	nfralxy 2414	. . . . 5 |- F/ x A. c e. A ( a R b -> ( a R c \/ c R b ) )
14	3, 13	nfralxy 2414	. . . 4 |- F/ x A. b e. A A. c e. A ( a R b -> ( a R c \/ c R b ) )
15	3, 14	nfralxy 2414	. . 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 1502	. 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 1408	1 |- F/ x R Or A

Syntax hints:   -> wi 4   /\ wa 102   \/ wo 664   F/wnf 1394   F/_wnfc 2215   A.wral 2359   class class class wbr 3837   Po wpo 4112   Or wor 4113

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-un 3001  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-po 4114  df-iso 4115

This theorem is referenced by: (None)