Theorem iota1 5347

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 2089	. 2 |- ( E! x ph <-> E. z A. x ( ph <-> x = z ) )
2		sp 1564	. . . . 5 |- ( A. x ( ph <-> x = z ) -> ( ph <-> x = z ) )
3		iotaval 5344	. . . . . 6 |- ( A. x ( ph <-> x = z ) -> ( iota x ph ) = z )
4	3	eqeq2d 2250	. . . . 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 2240	. . . 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 1651	. 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 1400   = wceq 1402   E.wex 1545   E!weu 2086   iotacio 5330

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-sn 3711  df-pr 3712  df-uni 3931  df-iota 5332

This theorem is referenced by:  iota2df  5358  sniota  5363  tz6.12-1  5717  riota1  6048  riota1a  6049  erovlem  6891