Theorem nfpo 4444

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 4439	. 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 2392	. . . . . . . 8 |- F/_ x a
4		nfpo.r	. . . . . . . 8 |- F/_ x R
5	3, 4, 3	nfbr 4175	. . . . . . 7 |- F/ x a R a
6	5	nfn 1710	. . . . . 6 |- F/ x -. a R a
7		nfcv 2392	. . . . . . . . 9 |- F/_ x b
8	3, 4, 7	nfbr 4175	. . . . . . . 8 |- F/ x a R b
9		nfcv 2392	. . . . . . . . 9 |- F/_ x c
10	7, 4, 9	nfbr 4175	. . . . . . . 8 |- F/ x b R c
11	8, 10	nfan 1618	. . . . . . 7 |- F/ x ( a R b /\ b R c )
12	3, 4, 9	nfbr 4175	. . . . . . 7 |- F/ x a R c
13	11, 12	nfim 1625	. . . . . 6 |- F/ x ( ( a R b /\ b R c ) -> a R c )
14	6, 13	nfan 1618	. . . . 5 |- F/ x ( -. a R a /\ ( ( a R b /\ b R c ) -> a R c ) )
15	2, 14	nfralxy 2588	. . . 4 |- F/ x A. c e. A ( -. a R a /\ ( ( a R b /\ b R c ) -> a R c ) )
16	2, 15	nfralxy 2588	. . 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 2588	. 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 1527	1 |- F/ x R Po A

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   F/wnf 1513   F/_wnfc 2379   A.wral 2528   class class class wbr 4128   Po wpo 4437

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-un 3224  df-sn 3714  df-pr 3715  df-op 3717  df-br 4129  df-po 4439

This theorem is referenced by:  nfso  4445