Theorem iota4 5114

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 2003	. 2 |- ( E! x ph <-> E. z A. x ( ph <-> x = z ) )
2		bi2 129	. . . . . 6 |- ( ( ph <-> x = z ) -> ( x = z -> ph ) )
3	2	alimi 1432	. . . . 5 |- ( A. x ( ph <-> x = z ) -> A. x ( x = z -> ph ) )
4		sb2 1741	. . . . 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 5107	. . . . . 6 |- ( A. x ( ph <-> x = z ) -> ( iota x ph ) = z )
7	6	eqcomd 2146	. . . . 5 |- ( A. x ( ph <-> x = z ) -> z = ( iota x ph ) )
8		dfsbcq2 2916	. . . . 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 146	. . 3 |- ( A. x ( ph <-> x = z ) -> [. ( iota x ph ) / x ]. ph )
11	10	exlimiv 1578	. 2 |- ( E. z A. x ( ph <-> x = z ) -> [. ( iota x ph ) / x ]. ph )
12	1, 11	sylbi 120	1 |- ( E! x ph -> [. ( iota x ph ) / x ]. ph )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1330   = wceq 1332   E.wex 1469   [wsb 1736   E!weu 2000   [.wsbc 2913   iotacio 5094

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2691  df-sbc 2914  df-un 3080  df-sn 3538  df-pr 3539  df-uni 3745  df-iota 5096

This theorem is referenced by:  iota4an  5115  iotacl  5119