2eu3 - Metamath Proof Explorer

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Theorem 2eu3 2555

Description: Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)

**Assertion**
Ref	Expression
2eu3	$\/$ $/\$ $/\$

**Proof of Theorem 2eu3**
Step	Hyp	Ref	Expression
1		nfmo1 2481	. . . . 5
2	1	19.31 2102	. . . 4 $\/$ $\/$
3	2	albii 1747	. . 3 $\/$ $\/$
4		nfmo1 2481	. . . . 5
5	4	nfal 2153	. . . 4
6	5	19.32 2101	. . 3 $\/$ $\/$
7	3, 6	bitri 264	. 2 $\/$ $\/$
8		2eu1 2553	. . . . . . 7 $/\$
9	8	biimpd 219	. . . . . 6 $/\$
10		ancom 466	. . . . . 6 $/\$ $/\$
11	9, 10	syl6ib 241	. . . . 5 $/\$
12	11	adantld 483	. . . 4 $/\$ $/\$
13		2eu1 2553	. . . . . 6 $/\$
14	13	biimpd 219	. . . . 5 $/\$
15	14	adantrd 484	. . . 4 $/\$ $/\$
16	12, 15	jaoi 394	. . 3 $\/$ $/\$ $/\$
17		2exeu 2549	. . . 4 $/\$
18		2exeu 2549	. . . . 5 $/\$
19	18	ancoms 469	. . . 4 $/\$
20	17, 19	jca 554	. . 3 $/\$ $/\$
21	16, 20	impbid1 215	. 2 $\/$ $/\$ $/\$
22	7, 21	sylbi 207	1 $\/$ $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $\/$ wo 383 $/\$ wa 384

wal 1481

wex 1704

weu 2470

wmo 2471

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-eu 2474 df-mo 2475

This theorem is referenced by: (None)