Theorem nfso 4039

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 4034	. 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 4038	. . 3 |- F/ x R Po A
5		nfcv 2178	. . . . . . . 8 |- F/_ x a
6		nfcv 2178	. . . . . . . 8 |- F/_ x b
7	5, 2, 6	nfbr 3808	. . . . . . 7 |- F/ x a R b
8		nfcv 2178	. . . . . . . . 9 |- F/_ x c
9	5, 2, 8	nfbr 3808	. . . . . . . 8 |- F/ x a R c
10	8, 2, 6	nfbr 3808	. . . . . . . 8 |- F/ x c R b
11	9, 10	nfor 1466	. . . . . . 7 |- F/ x ( a R c \/ c R b )
12	7, 11	nfim 1464	. . . . . 6 |- F/ x ( a R b -> ( a R c \/ c R b ) )
13	3, 12	nfralxy 2360	. . . . 5 |- F/ x A. c e. A ( a R b -> ( a R c \/ c R b ) )
14	3, 13	nfralxy 2360	. . . 4 |- F/ x A. b e. A A. c e. A ( a R b -> ( a R c \/ c R b ) )
15	3, 14	nfralxy 2360	. . 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 1457	. 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 1363	1 |- F/ x R Or A

Syntax hints:   -> wi 4   /\ wa 97   \/ wo 629   F/wnf 1349   F/_wnfc 2165   A.wral 2306   class class class wbr 3764   Po wpo 4031   Or wor 4032

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-po 4033  df-iso 4034

This theorem is referenced by: (None)