Theorem nfso 5041

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 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-so 5036	. 2 |- ( R Or A <-> ( R Po A /\ A. a e. A A. b e. A ( a R b \/ a = b \/ b R a ) ) )
2		nfpo.r	. . . 4 |- F/_ x R
3		nfpo.a	. . . 4 |- F/_ x A
4	2, 3	nfpo 5040	. . 3 |- F/ x R Po A
5		nfcv 2764	. . . . . . 7 |- F/_ x a
6		nfcv 2764	. . . . . . 7 |- F/_ x b
7	5, 2, 6	nfbr 4699	. . . . . 6 |- F/ x a R b
8		nfv 1843	. . . . . 6 |- F/ x a = b
9	6, 2, 5	nfbr 4699	. . . . . 6 |- F/ x b R a
10	7, 8, 9	nf3or 1835	. . . . 5 |- F/ x ( a R b \/ a = b \/ b R a )
11	3, 10	nfral 2945	. . . 4 |- F/ x A. b e. A ( a R b \/ a = b \/ b R a )
12	3, 11	nfral 2945	. . 3 |- F/ x A. a e. A A. b e. A ( a R b \/ a = b \/ b R a )
13	4, 12	nfan 1828	. 2 |- F/ x ( R Po A /\ A. a e. A A. b e. A ( a R b \/ a = b \/ b R a ) )
14	1, 13	nfxfr 1779	1 |- F/ x R Or A

Syntax hints:   /\ wa 384   \/ w3o 1036   F/wnf 1708   F/_wnfc 2751   A.wral 2912   class class class wbr 4653   Po wpo 5033   Or wor 5034

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-po 5035  df-so 5036

This theorem is referenced by:  nfwe  5090