Theorem nfpo 4038

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 4033	. 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 2178	. . . . . . . 8 |- F/_ x a
4		nfpo.r	. . . . . . . 8 |- F/_ x R
5	3, 4, 3	nfbr 3808	. . . . . . 7 |- F/ x a R a
6	5	nfn 1548	. . . . . 6 |- F/ x -. a R a
7		nfcv 2178	. . . . . . . . 9 |- F/_ x b
8	3, 4, 7	nfbr 3808	. . . . . . . 8 |- F/ x a R b
9		nfcv 2178	. . . . . . . . 9 |- F/_ x c
10	7, 4, 9	nfbr 3808	. . . . . . . 8 |- F/ x b R c
11	8, 10	nfan 1457	. . . . . . 7 |- F/ x ( a R b /\ b R c )
12	3, 4, 9	nfbr 3808	. . . . . . 7 |- F/ x a R c
13	11, 12	nfim 1464	. . . . . 6 |- F/ x ( ( a R b /\ b R c ) -> a R c )
14	6, 13	nfan 1457	. . . . 5 |- F/ x ( -. a R a /\ ( ( a R b /\ b R c ) -> a R c ) )
15	2, 14	nfralxy 2360	. . . 4 |- F/ x A. c e. A ( -. a R a /\ ( ( a R b /\ b R c ) -> a R c ) )
16	2, 15	nfralxy 2360	. . 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 2360	. 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 1363	1 |- F/ x R Po A

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   F/wnf 1349   F/_wnfc 2165   A.wral 2306   class class class wbr 3764   Po wpo 4031

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-po 4033

This theorem is referenced by:  nfso  4039