Theorem iotabi 5094

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

Assertion

Ref	Expression
iotabi	|- (A.x(ph <-> ps) -> (iotaxph) = (iotaxps))

Proof of Theorem iotabi

Step	Hyp	Ref	Expression
1		eq2ab 2004	. . . . . 6 |- ({x | ph} = {x | ps} <-> A.x(ph <-> ps))
2	1	biimpri 169	. . . . 5 |- (A.x(ph <-> ps) -> {x | ph} = {x | ps})
3	2	eqeq1d 1892	. . . 4 |- (A.x(ph <-> ps) -> ({x | ph} = {z} <-> {x | ps} = {z}))
4	3	abbidv 2008	. . 3 |- (A.x(ph <-> ps) -> {z | {x | ph} = {z}} = {z | {x | ps} = {z}})
5	4	unieqd 3188	. 2 |- (A.x(ph <-> ps) -> U.{z | {x | ph} = {z}} = U.{z | {x | ps} = {z}})
6		df-iota 5089	. 2 |- (iotaxph) = U.{z | {x | ph} = {z}}
7		df-iota 5089	. 2 |- (iotaxps) = U.{z | {x | ps} = {z}}
8	5, 6, 7	3eqtr4g 1953	1 |- (A.x(ph <-> ps) -> (iotaxph) = (iotaxps))

Syntax hints:   -> wi 3   <-> wb 163  A.wal 1296   = wceq 1298  {cab 1871  {csn 3044  U.cuni 3177  iotacio 5087

This theorem is referenced by:  iotabidv 5102  reiota4 5107  riotav 5565  iotasbcq 16403  stbbi 16726

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-rex 2110  df-uni 3178  df-iota 5089

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