Theorem reu2eqd 3296

Description: Deduce equality from restricted uniqueness, deduction version. (Contributed by Thierry Arnoux, 27-Nov-2019.)

Hypotheses

Ref	Expression
reu2eqd.1
reu2eqd.2
reu2eqd.3
reu2eqd.4
reu2eqd.5
reu2eqd.6
reu2eqd.7

Assertion

Ref	Expression
reu2eqd

Distinct variable groups:   ,   ,   ,   ,   ,

Proof of Theorem reu2eqd

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		reu2eqd.6	. 2
2		reu2eqd.7	. 2
3		reu2eqd.3	. . . . 5
4		reu2 3287	. . . . 5
5	3, 4	sylib 196	. . . 4
6	5	simprd 463	. . 3
7		reu2eqd.4	. . . 4
8		reu2eqd.5	. . . 4
9		nfv 1707	. . . . . . 7
10		nfs1v 2181	. . . . . . 7
11	9, 10	nfan 1928	. . . . . 6
12		nfv 1707	. . . . . 6
13	11, 12	nfim 1920	. . . . 5
14		nfv 1707	. . . . 5
15		reu2eqd.1	. . . . . . 7
16	15	anbi1d 704	. . . . . 6
17		eqeq1 2461	. . . . . 6
18	16, 17	imbi12d 320	. . . . 5
19		nfv 1707	. . . . . . . 8
20		reu2eqd.2	. . . . . . . 8
21	19, 20	sbhypf 3156	. . . . . . 7
22	21	anbi2d 703	. . . . . 6
23		eqeq2 2472	. . . . . 6
24	22, 23	imbi12d 320	. . . . 5
25	13, 14, 18, 24	rspc2 3218	. . . 4
26	7, 8, 25	syl2anc 661	. . 3
27	6, 26	mpd 15	. 2
28	1, 2, 27	mp2and 679	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  [wsb 1739  e.wcel 1818  A.wral 2807  E.wrex 2808  E!wreu 2809

This theorem is referenced by:  qtophmeo  20318  footeq  24098  mideulem2  24108  lmieq  24157

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-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-reu 2814  df-v 3111