Theorem iota4 5298

Description: Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.)

Assertion

Ref	Expression
iota4	|- ( E! x ph -> [. ( iota x ph ) / x ]. ph )

Proof of Theorem iota4

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		biimpr 130	. . . . . 6 |- ( ( ph <-> x = z ) -> ( x = z -> ph ) )
3	2	alimi 1501	. . . . 5 |- ( A. x ( ph <-> x = z ) -> A. x ( x = z -> ph ) )
4		sb2 1813	. . . . 5 |- ( A. x ( x = z -> ph ) -> [ z / x ] ph )
5	3, 4	syl 14	. . . 4 |- ( A. x ( ph <-> x = z ) -> [ z / x ] ph )
6		iotaval 5290	. . . . . 6 |- ( A. x ( ph <-> x = z ) -> ( iota x ph ) = z )
7	6	eqcomd 2235	. . . . 5 |- ( A. x ( ph <-> x = z ) -> z = ( iota x ph ) )
8		dfsbcq2 3031	. . . . 5 |- ( z = ( iota x ph ) -> ( [ z / x ] ph <-> [. ( iota x ph ) / x ]. ph ) )
9	7, 8	syl 14	. . . 4 |- ( A. x ( ph <-> x = z ) -> ( [ z / x ] ph <-> [. ( iota x ph ) / x ]. ph ) )
10	5, 9	mpbid 147	. . 3 |- ( A. x ( ph <-> x = z ) -> [. ( iota x ph ) / x ]. ph )
11	10	exlimiv 1644	. 2 |- ( E. z A. x ( ph <-> x = z ) -> [. ( iota x ph ) / x ]. ph )
12	1, 11	sylbi 121	1 |- ( E! x ph -> [. ( iota x ph ) / x ]. ph )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1393   = wceq 1395   E.wex 1538   [wsb 1808   E!weu 2077   [.wsbc 3028   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:  iota4an  5299  iotacl  5303