Theorem nfpo 4056

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 4051	. 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 2219	. . . . . . . 8 |- F/_ x a
4		nfpo.r	. . . . . . . 8 |- F/_ x R
5	3, 4, 3	nfbr 3829	. . . . . . 7 |- F/ x a R a
6	5	nfn 1588	. . . . . 6 |- F/ x -. a R a
7		nfcv 2219	. . . . . . . . 9 |- F/_ x b
8	3, 4, 7	nfbr 3829	. . . . . . . 8 |- F/ x a R b
9		nfcv 2219	. . . . . . . . 9 |- F/_ x c
10	7, 4, 9	nfbr 3829	. . . . . . . 8 |- F/ x b R c
11	8, 10	nfan 1497	. . . . . . 7 |- F/ x ( a R b /\ b R c )
12	3, 4, 9	nfbr 3829	. . . . . . 7 |- F/ x a R c
13	11, 12	nfim 1504	. . . . . 6 |- F/ x ( ( a R b /\ b R c ) -> a R c )
14	6, 13	nfan 1497	. . . . 5 |- F/ x ( -. a R a /\ ( ( a R b /\ b R c ) -> a R c ) )
15	2, 14	nfralxy 2402	. . . 4 |- F/ x A. c e. A ( -. a R a /\ ( ( a R b /\ b R c ) -> a R c ) )
16	2, 15	nfralxy 2402	. . 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 2402	. 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 1403	1 |- F/ x R Po A

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   F/wnf 1389   F/_wnfc 2206   A.wral 2348   class class class wbr 3785   Po wpo 4049

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-po 4051

This theorem is referenced by:  nfso  4057