2exeu - Intuitionistic Logic Explorer

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Theorem 2exeu 2111

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 1657	. . . . 5
2		hbe1 1488	. . . . . . . 8
3	2	hbmo 2058	. . . . . . 7
4	3	19.41h 1678	. . . . . 6 $/\$ $/\$
5		19.8a 1583	. . . . . . . . 9
6	5	moimi 2084	. . . . . . . 8
7	6	anim2i 340	. . . . . . 7 $/\$ $/\$
8	7	eximi 1593	. . . . . 6 $/\$ $/\$
9	4, 8	sylbir 134	. . . . 5 $/\$ $/\$
10	1, 9	sylanb 282	. . . 4 $/\$ $/\$
11		simpl 108	. . . . . 6 $/\$
12	11	moimi 2084	. . . . 5 $/\$
13	12	adantl 275	. . . 4 $/\$ $/\$
14	10, 13	anim12i 336	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
15	14	ancoms 266	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
16		eu5 2066	. . 3 $/\$
17		eu5 2066	. . 3 $/\$
18	16, 17	anbi12i 457	. 2 $/\$ $/\$ $/\$ $/\$
19		eu5 2066	. . 3 $/\$
20		eu5 2066	. . . . 5 $/\$
21	20	exbii 1598	. . . 4 $/\$
22	20	mobii 2056	. . . 4 $/\$
23	21, 22	anbi12i 457	. . 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 1485

weu 2019

wmo 2020

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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528

This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023

This theorem is referenced by: (None)