Theorem nfpo 4348

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 4343	. 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 2348	. . . . . . . 8 |- F/_ x a
4		nfpo.r	. . . . . . . 8 |- F/_ x R
5	3, 4, 3	nfbr 4090	. . . . . . 7 |- F/ x a R a
6	5	nfn 1681	. . . . . 6 |- F/ x -. a R a
7		nfcv 2348	. . . . . . . . 9 |- F/_ x b
8	3, 4, 7	nfbr 4090	. . . . . . . 8 |- F/ x a R b
9		nfcv 2348	. . . . . . . . 9 |- F/_ x c
10	7, 4, 9	nfbr 4090	. . . . . . . 8 |- F/ x b R c
11	8, 10	nfan 1588	. . . . . . 7 |- F/ x ( a R b /\ b R c )
12	3, 4, 9	nfbr 4090	. . . . . . 7 |- F/ x a R c
13	11, 12	nfim 1595	. . . . . 6 |- F/ x ( ( a R b /\ b R c ) -> a R c )
14	6, 13	nfan 1588	. . . . 5 |- F/ x ( -. a R a /\ ( ( a R b /\ b R c ) -> a R c ) )
15	2, 14	nfralxy 2544	. . . 4 |- F/ x A. c e. A ( -. a R a /\ ( ( a R b /\ b R c ) -> a R c ) )
16	2, 15	nfralxy 2544	. . 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 2544	. 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 1497	1 |- F/ x R Po A

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   F/wnf 1483   F/_wnfc 2335   A.wral 2484   class class class wbr 4044   Po wpo 4341

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-po 4343

This theorem is referenced by:  nfso  4349