Theorem iotaeq 4347

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

Assertion

Ref	Expression
iotaeq	|- (A.x x = y -> (iotaxph) = (iotayph))

Proof of Theorem iotaeq

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		drsb1 2022	. . . . . . 7 |- (A.x x = y -> ([z / x]ph <-> [z / y]ph))
2		df-clab 2340	. . . . . . 7 |- (z e. {x | ph} <-> [z / x]ph)
3		df-clab 2340	. . . . . . 7 |- (z e. {y | ph} <-> [z / y]ph)
4	1, 2, 3	3bitr4g 279	. . . . . 6 |- (A.x x = y -> (z e. {x | ph} <-> z e. {y | ph}))
5	4	eqrdv 2351	. . . . 5 |- (A.x x = y -> {x | ph} = {y | ph})
6	5	eqeq1d 2361	. . . 4 |- (A.x x = y -> ({x | ph} = {z} <-> {y | ph} = {z}))
7	6	abbidv 2467	. . 3 |- (A.x x = y -> {z | {x | ph} = {z}} = {z | {y | ph} = {z}})
8	7	unieqd 3902	. 2 |- (A.x x = y -> U.{z | {x | ph} = {z}} = U.{z | {y | ph} = {z}})
9		df-iota 4339	. 2 |- (iotaxph) = U.{z | {x | ph} = {z}}
10		df-iota 4339	. 2 |- (iotayph) = U.{z | {y | ph} = {z}}
11	8, 9, 10	3eqtr4g 2410	1 |- (A.x x = y -> (iotaxph) = (iotayph))

Syntax hints:   -> wi 4  A.wal 1540   = wceq 1642  [wsb 1648   e. wcel 1710  {cab 2339  {csn 3737  U.cuni 3891  iotacio 4337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-uni 3892  df-iota 4339

This theorem is referenced by: (None)