2exeu - Intuitionistic Logic Explorer

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Theorem 2exeu 2179

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

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

**Proof of Theorem 2exeu**
Step	Hyp	Ref	Expression
1		excom 1716	. . . . 5
2		hbe1 1548	. . . . . . . 8
3	2	hbmo 2125	. . . . . . 7
4	3	19.41h 1737	. . . . . 6 $/\$ $/\$
5		19.8a 1643	. . . . . . . . 9
6	5	moimi 2152	. . . . . . . 8
7	6	anim2i 342	. . . . . . 7 $/\$ $/\$
8	7	eximi 1653	. . . . . 6 $/\$ $/\$
9	4, 8	sylbir 135	. . . . 5 $/\$ $/\$
10	1, 9	sylanb 284	. . . 4 $/\$ $/\$
11		simpl 109	. . . . . 6 $/\$
12	11	moimi 2152	. . . . 5 $/\$
13	12	adantl 277	. . . 4 $/\$ $/\$
14	10, 13	anim12i 338	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
15	14	ancoms 268	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
16		eu5 2134	. . 3 $/\$
17		eu5 2134	. . 3 $/\$
18	16, 17	anbi12i 464	. 2 $/\$ $/\$ $/\$ $/\$
19		eu5 2134	. . 3 $/\$
20		eu5 2134	. . . . 5 $/\$
21	20	exbii 1658	. . . 4 $/\$
22	20	mobii 2123	. . . 4 $/\$
23	21, 22	anbi12i 464	. . 3 $/\$ $/\$ $/\$ $/\$
24	19, 23	bitri 184	. 2 $/\$ $/\$ $/\$
25	15, 18, 24	3imtr4i 201	1 $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wex 1545

weu 2086

wmo 2087

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588

This theorem depends on definitions: df-bi 117 df-tru 1405 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090

This theorem is referenced by: (None)