2eu8 - Metamath Proof Explorer

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Theorem 2eu8 2560

Description: Two equivalent expressions for double existential uniqueness. Curiously, we can put

on either of the internal conjuncts but not both. We can also commute

using 2eu7 2559. (Contributed by NM, 20-Feb-2005.)

**Assertion**
Ref	Expression
2eu8	$/\$ $/\$

**Proof of Theorem 2eu8**
Step	Hyp	Ref	Expression
1		2eu2 2554	. . 3
2	1	pm5.32i 669	. 2 $/\$ $/\$
3		nfeu1 2480	. . . . 5
4	3	nfeu 2486	. . . 4
5	4	euan 2530	. . 3 $/\$ $/\$
6		ancom 466	. . . . . 6 $/\$ $/\$
7	6	eubii 2492	. . . . 5 $/\$ $/\$
8		nfe1 2027	. . . . . 6
9	8	euan 2530	. . . . 5 $/\$ $/\$
10		ancom 466	. . . . 5 $/\$ $/\$
11	7, 9, 10	3bitri 286	. . . 4 $/\$ $/\$
12	11	eubii 2492	. . 3 $/\$ $/\$
13		ancom 466	. . 3 $/\$ $/\$
14	5, 12, 13	3bitr4ri 293	. 2 $/\$ $/\$
15		2eu7 2559	. 2 $/\$ $/\$
16	2, 14, 15	3bitr3ri 291	1 $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wb 196 $/\$ wa 384

wex 1704

weu 2470

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)