Theorem iota1 5167

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 2017	. 2 |- ( E! x ph <-> E. z A. x ( ph <-> x = z ) )
2		sp 1499	. . . . 5 |- ( A. x ( ph <-> x = z ) -> ( ph <-> x = z ) )
3		iotaval 5164	. . . . . 6 |- ( A. x ( ph <-> x = z ) -> ( iota x ph ) = z )
4	3	eqeq2d 2177	. . . . 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 2167	. . . 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 1586	. 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 1341   = wceq 1343   E.wex 1480   E!weu 2014   iotacio 5151

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-sn 3582  df-pr 3583  df-uni 3790  df-iota 5153

This theorem is referenced by:  iota2df  5177  sniota  5180  tz6.12-1  5513  riota1  5816  riota1a  5817  erovlem  6593