2reu3 - Mathbox for Alexander van der Vekens

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Theorem 2reu3 41188

Description: Double restricted existential uniqueness, analogous to 2eu3 2555. (Contributed by Alexander van der Vekens, 29-Jun-2017.)

**Assertion**
Ref	Expression
2reu3	$\/$ $/\$ $/\$

(

)

**Proof of Theorem 2reu3**
Step	Hyp	Ref	Expression
1		orcom 402	. . . . . . 7 $\/$ $\/$
2	1	ralbii 2980	. . . . . 6 $\/$ $\/$
3		nfrmo1 3111	. . . . . . 7
4	3	r19.32 41167	. . . . . 6 $\/$ $\/$
5	2, 4	bitri 264	. . . . 5 $\/$ $\/$
6		orcom 402	. . . . 5 $\/$ $\/$
7	5, 6	bitri 264	. . . 4 $\/$ $\/$
8	7	ralbii 2980	. . 3 $\/$ $\/$
9		nfcv 2764	. . . . 5
10		nfrmo1 3111	. . . . 5
11	9, 10	nfral 2945	. . . 4
12	11	r19.32 41167	. . 3 $\/$ $\/$
13	8, 12	bitri 264	. 2 $\/$ $\/$
14		2reu1 41186	. . . . . . 7 $/\$
15	14	biimpd 219	. . . . . 6 $/\$
16		ancom 466	. . . . . 6 $/\$ $/\$
17	15, 16	syl6ib 241	. . . . 5 $/\$
18	17	adantld 483	. . . 4 $/\$ $/\$
19		2reu1 41186	. . . . . 6 $/\$
20	19	biimpd 219	. . . . 5 $/\$
21	20	adantrd 484	. . . 4 $/\$ $/\$
22	18, 21	jaoi 394	. . 3 $\/$ $/\$ $/\$
23		2rexreu 41185	. . . 4 $/\$
24		2rexreu 41185	. . . . 5 $/\$
25	24	ancoms 469	. . . 4 $/\$
26	23, 25	jca 554	. . 3 $/\$ $/\$
27	22, 26	impbid1 215	. 2 $\/$ $/\$ $/\$
28	13, 27	sylbi 207	1 $\/$ $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $\/$ wo 383 $/\$ wa 384

wral 2912

wrex 2913

wreu 2914

wrmo 2915

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-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920

This theorem is referenced by: (None)