Theorem nfpo 4303

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 4298	. 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 2319	. . . . . . . 8 |- F/_ x a
4		nfpo.r	. . . . . . . 8 |- F/_ x R
5	3, 4, 3	nfbr 4051	. . . . . . 7 |- F/ x a R a
6	5	nfn 1658	. . . . . 6 |- F/ x -. a R a
7		nfcv 2319	. . . . . . . . 9 |- F/_ x b
8	3, 4, 7	nfbr 4051	. . . . . . . 8 |- F/ x a R b
9		nfcv 2319	. . . . . . . . 9 |- F/_ x c
10	7, 4, 9	nfbr 4051	. . . . . . . 8 |- F/ x b R c
11	8, 10	nfan 1565	. . . . . . 7 |- F/ x ( a R b /\ b R c )
12	3, 4, 9	nfbr 4051	. . . . . . 7 |- F/ x a R c
13	11, 12	nfim 1572	. . . . . 6 |- F/ x ( ( a R b /\ b R c ) -> a R c )
14	6, 13	nfan 1565	. . . . 5 |- F/ x ( -. a R a /\ ( ( a R b /\ b R c ) -> a R c ) )
15	2, 14	nfralxy 2515	. . . 4 |- F/ x A. c e. A ( -. a R a /\ ( ( a R b /\ b R c ) -> a R c ) )
16	2, 15	nfralxy 2515	. . 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 2515	. 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 1474	1 |- F/ x R Po A

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   F/wnf 1460   F/_wnfc 2306   A.wral 2455   class class class wbr 4005   Po wpo 4296

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-po 4298

This theorem is referenced by:  nfso  4304