Theorem iota1 5246

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 2057	. 2 |- ( E! x ph <-> E. z A. x ( ph <-> x = z ) )
2		sp 1534	. . . . 5 |- ( A. x ( ph <-> x = z ) -> ( ph <-> x = z ) )
3		iotaval 5243	. . . . . 6 |- ( A. x ( ph <-> x = z ) -> ( iota x ph ) = z )
4	3	eqeq2d 2217	. . . . 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 2207	. . . 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 1621	. 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 1371   = wceq 1373   E.wex 1515   E!weu 2054   iotacio 5230

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-sn 3639  df-pr 3640  df-uni 3851  df-iota 5232

This theorem is referenced by:  iota2df  5257  sniota  5262  tz6.12-1  5603  riota1  5918  riota1a  5919  erovlem  6714