Theorem iota1 5293

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 2080	. 2 |- ( E! x ph <-> E. z A. x ( ph <-> x = z ) )
2		sp 1557	. . . . 5 |- ( A. x ( ph <-> x = z ) -> ( ph <-> x = z ) )
3		iotaval 5290	. . . . . 6 |- ( A. x ( ph <-> x = z ) -> ( iota x ph ) = z )
4	3	eqeq2d 2241	. . . . 5 |- ( A. x ( ph <-> x = z ) -> ( x = ( iota x ph ) <-> x = z ) )
5	2, 4	bitr4d 191	. . . 4 |- ( A. x ( ph <-> x = z ) -> ( ph <-> x = ( iota x ph ) ) )
6		eqcom 2231	. . . 4 |- ( x = ( iota x ph ) <-> ( iota x ph ) = x )
7	5, 6	bitrdi 196	. . 3 |- ( A. x ( ph <-> x = z ) -> ( ph <-> ( iota x ph ) = x ) )
8	7	exlimiv 1644	. 2 |- ( E. z A. x ( ph <-> x = z ) -> ( ph <-> ( iota x ph ) = x ) )
9	1, 8	sylbi 121	1 |- ( E! x ph -> ( ph <-> ( iota x ph ) = x ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1393   = wceq 1395   E.wex 1538   E!weu 2077   iotacio 5276

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-sn 3672  df-pr 3673  df-uni 3889  df-iota 5278

This theorem is referenced by:  iota2df  5304  sniota  5309  tz6.12-1  5654  riota1  5974  riota1a  5975  erovlem  6774