Theorem nfse 4437

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 4429	. 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 2373	. . . . . 6 |- F/_ x a
4		nfse.r	. . . . . 6 |- F/_ x R
5		nfcv 2373	. . . . . 6 |- F/_ x b
6	3, 4, 5	nfbr 4134	. . . . 5 |- F/ x a R b
7	6, 2	nfrabw 2713	. . . 4 |- F/_ x { a e. A | a R b }
8	7	nfel1 2384	. . 3 |- F/ x { a e. A | a R b } e. _V
9	2, 8	nfralxy 2569	. 2 |- F/ x A. b e. A { a e. A | a R b } e. _V
10	1, 9	nfxfr 1522	1 |- F/ x R Se A

Syntax hints:   F/wnf 1508   e. wcel 2201   F/_wnfc 2360   A.wral 2509   {crab 2513   _Vcvv 2801   class class class wbr 4087   Se wse 4425

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2212

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rab 2518  df-v 2803  df-un 3203  df-sn 3674  df-pr 3675  df-op 3677  df-br 4088  df-se 4429

This theorem is referenced by: (None)