Theorem preqsn 4213

Description: Equivalence for a pair equal to a singleton. (Contributed by NM, 3-Jun-2008.)

Hypotheses

Ref	Expression
preqsn.1
preqsn.2
preqsn.3

Assertion

Ref	Expression
preqsn

Proof of Theorem preqsn

Step	Hyp	Ref	Expression
1		dfsn2 4042	. . 3
2	1	eqeq2i 2475	. 2
3		preqsn.1	. . . 4
4		preqsn.2	. . . 4
5		preqsn.3	. . . 4
6	3, 4, 5, 5	preq12b 4206	. . 3
7		oridm 514	. . . 4
8		eqtr3 2485	. . . . . 6
9		simpr 461	. . . . . 6
10	8, 9	jca 532	. . . . 5
11		eqtr 2483	. . . . . 6
12		simpr 461	. . . . . 6
13	11, 12	jca 532	. . . . 5
14	10, 13	impbii 188	. . . 4
15	7, 14	bitri 249	. . 3
16	6, 15	bitri 249	. 2
17	2, 16	bitri 249	1

Syntax hints:  <->wb 184  \/wo 368  /\wa 369  =wceq 1395  e.wcel 1818  cvv 3109  {csn 4029  {cpr 4031

This theorem is referenced by:  opeqsn  4748  relop  5158  hash2prde  12516  symg2bas  16423

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-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-un 3480  df-sn 4030  df-pr 4032