Theorem iota1 5174

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

Assertion

Ref	Expression
iota1	|- ( E! x ph -> ( ph <-> ( iota x ph ) = x ) )

Proof of Theorem iota1

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-eu 2022	. 2 |- ( E! x ph <-> E. z A. x ( ph <-> x = z ) )
2		sp 1504	. . . . 5 |- ( A. x ( ph <-> x = z ) -> ( ph <-> x = z ) )
3		iotaval 5171	. . . . . 6 |- ( A. x ( ph <-> x = z ) -> ( iota x ph ) = z )
4	3	eqeq2d 2182	. . . . 5 |- ( A. x ( ph <-> x = z ) -> ( x = ( iota x ph ) <-> x = z ) )
5	2, 4	bitr4d 190	. . . 4 |- ( A. x ( ph <-> x = z ) -> ( ph <-> x = ( iota x ph ) ) )
6		eqcom 2172	. . . 4 |- ( x = ( iota x ph ) <-> ( iota x ph ) = x )
7	5, 6	bitrdi 195	. . 3 |- ( A. x ( ph <-> x = z ) -> ( ph <-> ( iota x ph ) = x ) )
8	7	exlimiv 1591	. 2 |- ( E. z A. x ( ph <-> x = z ) -> ( ph <-> ( iota x ph ) = x ) )
9	1, 8	sylbi 120	1 |- ( E! x ph -> ( ph <-> ( iota x ph ) = x ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1346   = wceq 1348   E.wex 1485   E!weu 2019   iotacio 5158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-sn 3589  df-pr 3590  df-uni 3797  df-iota 5160

This theorem is referenced by:  iota2df  5184  sniota  5189  tz6.12-1  5523  riota1  5827  riota1a  5828  erovlem  6605