Theorem iota4 5106

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 2002	. 2 |- ( E! x ph <-> E. z A. x ( ph <-> x = z ) )
2		bi2 129	. . . . . 6 |- ( ( ph <-> x = z ) -> ( x = z -> ph ) )
3	2	alimi 1431	. . . . 5 |- ( A. x ( ph <-> x = z ) -> A. x ( x = z -> ph ) )
4		sb2 1740	. . . . 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 5099	. . . . . 6 |- ( A. x ( ph <-> x = z ) -> ( iota x ph ) = z )
7	6	eqcomd 2145	. . . . 5 |- ( A. x ( ph <-> x = z ) -> z = ( iota x ph ) )
8		dfsbcq2 2912	. . . . 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 1577	. 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 1329   = wceq 1331   E.wex 1468   [wsb 1735   E!weu 1999   [.wsbc 2909   iotacio 5086

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-sn 3533  df-pr 3534  df-uni 3737  df-iota 5088

This theorem is referenced by:  iota4an  5107  iotacl  5111