Theorem nfse 4372

Description: Bound-variable hypothesis builder for set-like relations. (Contributed by Mario Carneiro, 24-Jun-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)

Hypotheses

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

Assertion

Ref	Expression
nfse	|- F/ x R Se A

Proof of Theorem nfse

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

Step	Hyp	Ref	Expression
1		df-se 4364	. 2 |- ( R Se A <-> A. b e. A { a e. A | a R b } e. _V )
2		nfse.a	. . 3 |- F/_ x A
3		nfcv 2336	. . . . . 6 |- F/_ x a
4		nfse.r	. . . . . 6 |- F/_ x R
5		nfcv 2336	. . . . . 6 |- F/_ x b
6	3, 4, 5	nfbr 4075	. . . . 5 |- F/ x a R b
7	6, 2	nfrabw 2675	. . . 4 |- F/_ x { a e. A | a R b }
8	7	nfel1 2347	. . 3 |- F/ x { a e. A | a R b } e. _V
9	2, 8	nfralxy 2532	. 2 |- F/ x A. b e. A { a e. A | a R b } e. _V
10	1, 9	nfxfr 1485	1 |- F/ x R Se A

Syntax hints:   F/wnf 1471   e. wcel 2164   F/_wnfc 2323   A.wral 2472   {crab 2476   _Vcvv 2760   class class class wbr 4029   Se wse 4360

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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rab 2481  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-se 4364

This theorem is referenced by: (None)