Theorem iota1 5570

Description: Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)

Assertion

Ref	Expression
iota1

Proof of Theorem iota1

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-eu 2286	. 2
2		sp 1859	. . . . 5
3		iotaval 5567	. . . . . 6
4	3	eqeq2d 2471	. . . . 5
5	2, 4	bitr4d 256	. . . 4
6		eqcom 2466	. . . 4
7	5, 6	syl6bb 261	. . 3
8	7	exlimiv 1722	. 2
9	1, 8	sylbi 195	1

Syntax hints:  ->wi 4  <->wb 184  A.wal 1393  =wceq 1395  E.wex 1612  E!weu 2282  iotacio 5554

This theorem is referenced by:  iota2df  5580  sniota  5583  tz6.12-1  5887  opabiota  5936  riota1  6276  riota1a  6277  erovlem  7426  gsumval3OLD  16908  gsumval3lem2  16910  bnj1366  33888

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-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rex 2813  df-v 3111  df-sbc 3328  df-un 3480  df-sn 4030  df-pr 4032  df-uni 4250  df-iota 5556