Theorem nffr 5088

Description: Bound-variable hypothesis builder for well-founded relations. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)

Hypotheses

Ref	Expression
nffr.r	|- F/_ x R
nffr.a	|- F/_ x A

Assertion

Ref	Expression
nffr	|- F/ x R Fr A

Proof of Theorem nffr

Dummy variables a b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fr 5073	. 2 |- ( R Fr A <-> A. a ( ( a C_ A /\ a =/= (/) ) -> E. b e. a A. c e. a -. c R b ) )
2		nfcv 2764	. . . . . 6 |- F/_ x a
3		nffr.a	. . . . . 6 |- F/_ x A
4	2, 3	nfss 3596	. . . . 5 |- F/ x a C_ A
5		nfv 1843	. . . . 5 |- F/ x a =/= (/)
6	4, 5	nfan 1828	. . . 4 |- F/ x ( a C_ A /\ a =/= (/) )
7		nfcv 2764	. . . . . . . 8 |- F/_ x c
8		nffr.r	. . . . . . . 8 |- F/_ x R
9		nfcv 2764	. . . . . . . 8 |- F/_ x b
10	7, 8, 9	nfbr 4699	. . . . . . 7 |- F/ x c R b
11	10	nfn 1784	. . . . . 6 |- F/ x -. c R b
12	2, 11	nfral 2945	. . . . 5 |- F/ x A. c e. a -. c R b
13	2, 12	nfrex 3007	. . . 4 |- F/ x E. b e. a A. c e. a -. c R b
14	6, 13	nfim 1825	. . 3 |- F/ x ( ( a C_ A /\ a =/= (/) ) -> E. b e. a A. c e. a -. c R b )
15	14	nfal 2153	. 2 |- F/ x A. a ( ( a C_ A /\ a =/= (/) ) -> E. b e. a A. c e. a -. c R b )
16	1, 15	nfxfr 1779	1 |- F/ x R Fr A

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   A.wal 1481   F/wnf 1708   F/_wnfc 2751   =/= wne 2794   A.wral 2912   E.wrex 2913   C_ wss 3574   (/)c0 3915   class class class wbr 4653   Fr wfr 5070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-fr 5073

This theorem is referenced by:  nfwe  5090