Theorem iota4 5270

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 2058	. 2 |- ( E! x ph <-> E. z A. x ( ph <-> x = z ) )
2		biimpr 130	. . . . . 6 |- ( ( ph <-> x = z ) -> ( x = z -> ph ) )
3	2	alimi 1479	. . . . 5 |- ( A. x ( ph <-> x = z ) -> A. x ( x = z -> ph ) )
4		sb2 1791	. . . . 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 5262	. . . . . 6 |- ( A. x ( ph <-> x = z ) -> ( iota x ph ) = z )
7	6	eqcomd 2213	. . . . 5 |- ( A. x ( ph <-> x = z ) -> z = ( iota x ph ) )
8		dfsbcq2 3008	. . . . 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 1622	. 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 1371   = wceq 1373   E.wex 1516   [wsb 1786   E!weu 2055   [.wsbc 3005   iotacio 5249

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-sn 3649  df-pr 3650  df-uni 3865  df-iota 5251

This theorem is referenced by:  iota4an  5271  iotacl  5275