Theorem nfso 4442

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 4437	. 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 4441	. . 3 |- F/ x R Po A
5		nfcv 2392	. . . . . . . 8 |- F/_ x a
6		nfcv 2392	. . . . . . . 8 |- F/_ x b
7	5, 2, 6	nfbr 4172	. . . . . . 7 |- F/ x a R b
8		nfcv 2392	. . . . . . . . 9 |- F/_ x c
9	5, 2, 8	nfbr 4172	. . . . . . . 8 |- F/ x a R c
10	8, 2, 6	nfbr 4172	. . . . . . . 8 |- F/ x c R b
11	9, 10	nfor 1627	. . . . . . 7 |- F/ x ( a R c \/ c R b )
12	7, 11	nfim 1625	. . . . . 6 |- F/ x ( a R b -> ( a R c \/ c R b ) )
13	3, 12	nfralxy 2588	. . . . 5 |- F/ x A. c e. A ( a R b -> ( a R c \/ c R b ) )
14	3, 13	nfralxy 2588	. . . 4 |- F/ x A. b e. A A. c e. A ( a R b -> ( a R c \/ c R b ) )
15	3, 14	nfralxy 2588	. . 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 1618	. 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 1527	1 |- F/ x R Or A

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 720   F/wnf 1513   F/_wnfc 2379   A.wral 2528   class class class wbr 4125   Po wpo 4434   Or wor 4435

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-un 3224  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-po 4436  df-iso 4437

This theorem is referenced by: (None)