2exeu - Intuitionistic Logic Explorer

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Theorem 2exeu 2092

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 1643	. . . . 5
2		hbe1 1472	. . . . . . . 8
3	2	hbmo 2039	. . . . . . 7
4	3	19.41h 1664	. . . . . 6 $/\$ $/\$
5		19.8a 1570	. . . . . . . . 9
6	5	moimi 2065	. . . . . . . 8
7	6	anim2i 340	. . . . . . 7 $/\$ $/\$
8	7	eximi 1580	. . . . . 6 $/\$ $/\$
9	4, 8	sylbir 134	. . . . 5 $/\$ $/\$
10	1, 9	sylanb 282	. . . 4 $/\$ $/\$
11		simpl 108	. . . . . 6 $/\$
12	11	moimi 2065	. . . . 5 $/\$
13	12	adantl 275	. . . 4 $/\$ $/\$
14	10, 13	anim12i 336	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
15	14	ancoms 266	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
16		eu5 2047	. . 3 $/\$
17		eu5 2047	. . 3 $/\$
18	16, 17	anbi12i 456	. 2 $/\$ $/\$ $/\$ $/\$
19		eu5 2047	. . 3 $/\$
20		eu5 2047	. . . . 5 $/\$
21	20	exbii 1585	. . . 4 $/\$
22	20	mobii 2037	. . . 4 $/\$
23	21, 22	anbi12i 456	. . 3 $/\$ $/\$ $/\$ $/\$
24	19, 23	bitri 183	. 2 $/\$ $/\$ $/\$
25	15, 18, 24	3imtr4i 200	1 $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wex 1469

weu 2000

wmo 2001

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516

This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004

This theorem is referenced by: (None)