Theorem iota2df 5580

Description: A condition that allows us to represent "the unique element such that " with a class expression . (Contributed by NM, 30-Dec-2014.)

Hypotheses

Ref	Expression
iota2df.1
iota2df.2
iota2df.3
iota2df.4
iota2df.5
iota2df.6

Assertion

Ref	Expression
iota2df

Proof of Theorem iota2df

Step	Hyp	Ref	Expression
1		iota2df.1	. 2
2		iota2df.3	. . 3
3		simpr 461	. . . 4
4	3	eqeq2d 2471	. . 3
5	2, 4	bibi12d 321	. 2
6		iota2df.2	. . 3
7		iota1 5570	. . 3
8	6, 7	syl 16	. 2
9		iota2df.4	. 2
10		iota2df.6	. 2
11		iota2df.5	. . 3
12		nfiota1 5558	. . . . 5
13	12	a1i 11	. . . 4
14	13, 10	nfeqd 2626	. . 3
15	11, 14	nfbid 1933	. 2
16	1, 5, 8, 9, 10, 15	vtocldf 3158	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  F/wnf 1616  e.wcel 1818  E!weu 2282  F/_wnfc 2605  iotacio 5554

This theorem is referenced by:  iota2d  5581  iota2  5582  riota2df  6278  opiota  6859

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-sb 1740  df-eu 2286  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-v 3111  df-sbc 3328  df-un 3480  df-sn 4030  df-pr 4032  df-uni 4250  df-iota 5556