Theorem iota4 5582

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 2313	. 2 |- ( E! x ph <-> E. z A. x ( ph <-> x = z ) )
2		biimpr 203	. . . . . 6 |- ( ( ph <-> x = z ) -> ( x = z -> ph ) )
3	2	alimi 1694	. . . . 5 |- ( A. x ( ph <-> x = z ) -> A. x ( x = z -> ph ) )
4		sb2 2193	. . . . 5 |- ( A. x ( x = z -> ph ) -> [ z / x ] ph )
5	3, 4	syl 17	. . . 4 |- ( A. x ( ph <-> x = z ) -> [ z / x ] ph )
6		iotaval 5575	. . . . . 6 |- ( A. x ( ph <-> x = z ) -> ( iota x ph ) = z )
7	6	eqcomd 2467	. . . . 5 |- ( A. x ( ph <-> x = z ) -> z = ( iota x ph ) )
8		dfsbcq2 3281	. . . . 5 |- ( z = ( iota x ph ) -> ( [ z / x ] ph <-> [. ( iota x ph ) / x ]. ph ) )
9	7, 8	syl 17	. . . 4 |- ( A. x ( ph <-> x = z ) -> ( [ z / x ] ph <-> [. ( iota x ph ) / x ]. ph ) )
10	5, 9	mpbid 215	. . 3 |- ( A. x ( ph <-> x = z ) -> [. ( iota x ph ) / x ]. ph )
11	10	exlimiv 1786	. 2 |- ( E. z A. x ( ph <-> x = z ) -> [. ( iota x ph ) / x ]. ph )
12	1, 11	sylbi 200	1 |- ( E! x ph -> [. ( iota x ph ) / x ]. ph )

Syntax hints:   -> wi 4   <-> wb 189   A.wal 1452   = wceq 1454   E.wex 1673   [wsb 1807   E!weu 2309   [.wsbc 3278   iotacio 5562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-rex 2754  df-v 3058  df-sbc 3279  df-un 3420  df-sn 3980  df-pr 3982  df-uni 4212  df-iota 5564

This theorem is referenced by:  iota4an  5583  iotacl  5587  pm14.24  36826  sbiota1  36828