Theorem 2ralunsn 4238

Description: Double restricted quantification over the union of a set and a singleton, using implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012.)

Hypotheses

Ref	Expression
2ralunsn.1
2ralunsn.2
2ralunsn.3

Assertion

Ref	Expression
2ralunsn

Distinct variable groups:   ,   ,,   ,   ,   ,   ,

Proof of Theorem 2ralunsn

Step	Hyp	Ref	Expression
1		2ralunsn.2	. . . 4
2	1	ralunsn 4237	. . 3
3	2	ralbidv 2896	. 2
4		2ralunsn.1	. . . . . 6
5	4	ralbidv 2896	. . . . 5
6		2ralunsn.3	. . . . 5
7	5, 6	anbi12d 710	. . . 4
8	7	ralunsn 4237	. . 3
9		r19.26 2984	. . . 4
10	9	anbi1i 695	. . 3
11	8, 10	syl6bb 261	. 2
12	3, 11	bitrd 253	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  A.wral 2807  u.cun 3473  {csn 4029

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-v 3111  df-sbc 3328  df-un 3480  df-sn 4030