Theorem 2reu5 3308

Description: Double restricted existential uniqueness in terms of restricted existential quantification and restricted universal quantification, analogous to 2eu5 2382 and reu3 3289. (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Assertion

Ref	Expression
2reu5

Distinct variable groups:   ,,,,   ,   ,,,   ,,   ,   ,

Proof of Theorem 2reu5

Step	Hyp	Ref	Expression
1		r19.29r 2993	. . . . . . . 8
2		r19.29r 2993	. . . . . . . . 9
3	2	reximi 2925	. . . . . . . 8
4		pm3.35 587	. . . . . . . . . 10
5	4	reximi 2925	. . . . . . . . 9
6	5	reximi 2925	. . . . . . . 8
7		eleq1 2529	. . . . . . . . . . . . 13
8		eleq1 2529	. . . . . . . . . . . . 13
9	7, 8	bi2anan9 873	. . . . . . . . . . . 12
10	9	biimpac 486	. . . . . . . . . . 11
11	10	ancomd 451	. . . . . . . . . 10
12	11	ex 434	. . . . . . . . 9
13	12	rexlimivv 2954	. . . . . . . 8
14	1, 3, 6, 13	4syl 21	. . . . . . 7
15	14	ex 434	. . . . . 6
16	15	pm4.71rd 635	. . . . 5
17		anass 649	. . . . 5
18	16, 17	syl6bb 261	. . . 4
19	18	2exbidv 1716	. . 3
20	19	pm5.32i 637	. 2
21		2reu5lem3 3307	. 2
22		df-rex 2813	. . . 4
23		r19.42v 3012	. . . . . 6
24		df-rex 2813	. . . . . 6
25	23, 24	bitr3i 251	. . . . 5
26	25	exbii 1667	. . . 4
27	22, 26	bitri 249	. . 3
28	27	anbi2i 694	. 2
29	20, 21, 28	3bitr4i 277	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  E.wex 1612  e.wcel 1818  A.wral 2807  E.wrex 2808  E!wreu 2809  E*wrmo 2810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-eu 2286  df-mo 2287  df-cleq 2449  df-clel 2452  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815