Theorem 2eu1 2376

Description: Double existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 11-Nov-2019.)

Assertion

Ref	Expression
2eu1

Proof of Theorem 2eu1

Step	Hyp	Ref	Expression
1		2eu2ex 2368	. . . . 5
2		df-mo 2287	. . . . . . 7
3	2	albii 1640	. . . . . 6
4		euim 2344	. . . . . . 7
5	4	ex 434	. . . . . 6
6	3, 5	syl5bi 217	. . . . 5
7	1, 6	syl 16	. . . 4
8	7	pm2.43b 50	. . 3
9		2euswap 2370	. . . 4
10	8, 9	syld 44	. . 3
11	8, 10	jcad 533	. 2
12		2exeu 2371	. 2
13	11, 12	impbid1 203	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  E.wex 1612  E!weu 2282  E*wmo 2283

This theorem is referenced by:  2eu2  2378  2eu3  2379  2eu5  2382

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

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-eu 2286  df-mo 2287