Theorem iotabi 5565

Description: Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)

Assertion

Ref	Expression
iotabi

Proof of Theorem iotabi

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		abbi 2588	. . . . . 6
2	1	biimpi 194	. . . . 5
3	2	eqeq1d 2459	. . . 4
4	3	abbidv 2593	. . 3
5	4	unieqd 4259	. 2
6		df-iota 5556	. 2
7		df-iota 5556	. 2
8	5, 6, 7	3eqtr4g 2523	1

Syntax hints:  ->wi 4  <->wb 184  A.wal 1393  =wceq 1395  {cab 2442  {csn 4029  U.cuni 4249  iotacio 5554

This theorem is referenced by:  iotabidv  5577  iotabii  5578  eusvobj1  6290  iotasbcq  31344

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-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rex 2813  df-uni 4250  df-iota 5556