Theorem nfse 4388

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 4380	. 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 2348	. . . . . 6 |- F/_ x a
4		nfse.r	. . . . . 6 |- F/_ x R
5		nfcv 2348	. . . . . 6 |- F/_ x b
6	3, 4, 5	nfbr 4090	. . . . 5 |- F/ x a R b
7	6, 2	nfrabw 2687	. . . 4 |- F/_ x { a e. A | a R b }
8	7	nfel1 2359	. . 3 |- F/ x { a e. A | a R b } e. _V
9	2, 8	nfralxy 2544	. 2 |- F/ x A. b e. A { a e. A | a R b } e. _V
10	1, 9	nfxfr 1497	1 |- F/ x R Se A

Syntax hints:   F/wnf 1483   e. wcel 2176   F/_wnfc 2335   A.wral 2484   {crab 2488   _Vcvv 2772   class class class wbr 4044   Se wse 4376

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 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-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rab 2493  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-se 4380

This theorem is referenced by: (None)