Theorem nfpo 4326

Description: Bound-variable hypothesis builder for partial 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
nfpo	|- F/ x R Po A

Proof of Theorem nfpo

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

Step	Hyp	Ref	Expression
1		df-po 4321	. 2 |- ( R Po A <-> A. a e. A A. b e. A A. c e. A ( -. a R a /\ ( ( a R b /\ b R c ) -> a R c ) ) )
2		nfpo.a	. . 3 |- F/_ x A
3		nfcv 2332	. . . . . . . 8 |- F/_ x a
4		nfpo.r	. . . . . . . 8 |- F/_ x R
5	3, 4, 3	nfbr 4071	. . . . . . 7 |- F/ x a R a
6	5	nfn 1669	. . . . . 6 |- F/ x -. a R a
7		nfcv 2332	. . . . . . . . 9 |- F/_ x b
8	3, 4, 7	nfbr 4071	. . . . . . . 8 |- F/ x a R b
9		nfcv 2332	. . . . . . . . 9 |- F/_ x c
10	7, 4, 9	nfbr 4071	. . . . . . . 8 |- F/ x b R c
11	8, 10	nfan 1576	. . . . . . 7 |- F/ x ( a R b /\ b R c )
12	3, 4, 9	nfbr 4071	. . . . . . 7 |- F/ x a R c
13	11, 12	nfim 1583	. . . . . 6 |- F/ x ( ( a R b /\ b R c ) -> a R c )
14	6, 13	nfan 1576	. . . . 5 |- F/ x ( -. a R a /\ ( ( a R b /\ b R c ) -> a R c ) )
15	2, 14	nfralxy 2528	. . . 4 |- F/ x A. c e. A ( -. a R a /\ ( ( a R b /\ b R c ) -> a R c ) )
16	2, 15	nfralxy 2528	. . 3 |- F/ x A. b e. A A. c e. A ( -. a R a /\ ( ( a R b /\ b R c ) -> a R c ) )
17	2, 16	nfralxy 2528	. 2 |- F/ x A. a e. A A. b e. A A. c e. A ( -. a R a /\ ( ( a R b /\ b R c ) -> a R c ) )
18	1, 17	nfxfr 1485	1 |- F/ x R Po A

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   F/wnf 1471   F/_wnfc 2319   A.wral 2468   class class class wbr 4025   Po wpo 4319

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 2758  df-un 3152  df-sn 3620  df-pr 3621  df-op 3623  df-br 4026  df-po 4321

This theorem is referenced by:  nfso  4327