Theorem nfse 4859

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
nffr.r
nffr.a

Assertion

Ref	Expression
nfse

Proof of Theorem nfse

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

Step	Hyp	Ref	Expression
1		df-se 4844	. 2
2		nffr.a	. . 3
3		nfcv 2619	. . . . . 6
4		nffr.r	. . . . . 6
5		nfcv 2619	. . . . . 6
6	3, 4, 5	nfbr 4496	. . . . 5
7	6, 2	nfrab 3039	. . . 4
8	7	nfel1 2635	. . 3
9	2, 8	nfral 2843	. 2
10	1, 9	nfxfr 1645	1

Syntax hints:  F/wnf 1616  e.wcel 1818  F/_wnfc 2605  A.wral 2807  {crab 2811  cvv 3109   class class class wbr 4452  Sewse 4841

This theorem is referenced by:  nfoi  7960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-se 4844