2exeu - Intuitionistic Logic Explorer

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Theorem 2exeu 2040

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 1599	. . . . 5
2		hbe1 1429	. . . . . . . 8
3	2	hbmo 1987	. . . . . . 7
4	3	19.41h 1620	. . . . . 6 $/\$ $/\$
5		19.8a 1527	. . . . . . . . 9
6	5	moimi 2013	. . . . . . . 8
7	6	anim2i 334	. . . . . . 7 $/\$ $/\$
8	7	eximi 1536	. . . . . 6 $/\$ $/\$
9	4, 8	sylbir 133	. . . . 5 $/\$ $/\$
10	1, 9	sylanb 278	. . . 4 $/\$ $/\$
11		simpl 107	. . . . . 6 $/\$
12	11	moimi 2013	. . . . 5 $/\$
13	12	adantl 271	. . . 4 $/\$ $/\$
14	10, 13	anim12i 331	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
15	14	ancoms 264	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
16		eu5 1995	. . 3 $/\$
17		eu5 1995	. . 3 $/\$
18	16, 17	anbi12i 448	. 2 $/\$ $/\$ $/\$ $/\$
19		eu5 1995	. . 3 $/\$
20		eu5 1995	. . . . 5 $/\$
21	20	exbii 1541	. . . 4 $/\$
22	20	mobii 1985	. . . 4 $/\$
23	21, 22	anbi12i 448	. . 3 $/\$ $/\$ $/\$ $/\$
24	19, 23	bitri 182	. 2 $/\$ $/\$ $/\$
25	15, 18, 24	3imtr4i 199	1 $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 102

wex 1426

weu 1948

wmo 1949

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473

This theorem depends on definitions: df-bi 115 df-tru 1292 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952

This theorem is referenced by: (None)