Theorem sbiota1 29686

Description: Theorem *14.25 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 12-Jul-2011.)

Assertion

Ref	Expression
sbiota1	|- ( E! x ph -> ( A. x ( ph -> ps ) <-> [. ( iota x ph ) / x ]. ps ) )

Proof of Theorem sbiota1

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-eu 2257	. . . 4 |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )
2	1	biimpi 194	. . 3 |- ( E! x ph -> E. y A. x ( ph <-> x = y ) )
3		iota4 5398	. . 3 |- ( E! x ph -> [. ( iota x ph ) / x ]. ph )
4		iotaval 5391	. . . . . 6 |- ( A. x ( ph <-> x = y ) -> ( iota x ph ) = y )
5	4	eqcomd 2447	. . . . 5 |- ( A. x ( ph <-> x = y ) -> y = ( iota x ph ) )
6		spsbim 2086	. . . . . . . 8 |- ( A. x ( ph -> ps ) -> ( [ y / x ] ph -> [ y / x ] ps ) )
7		sbsbc 3189	. . . . . . . 8 |- ( [ y / x ] ph <-> [. y / x ]. ph )
8		sbsbc 3189	. . . . . . . 8 |- ( [ y / x ] ps <-> [. y / x ]. ps )
9	6, 7, 8	3imtr3g 269	. . . . . . 7 |- ( A. x ( ph -> ps ) -> ( [. y / x ]. ph -> [. y / x ]. ps ) )
10		dfsbcq 3187	. . . . . . . 8 |- ( y = ( iota x ph ) -> ( [. y / x ]. ph <-> [. ( iota x ph ) / x ]. ph ) )
11		dfsbcq 3187	. . . . . . . 8 |- ( y = ( iota x ph ) -> ( [. y / x ]. ps <-> [. ( iota x ph ) / x ]. ps ) )
12	10, 11	imbi12d 320	. . . . . . 7 |- ( y = ( iota x ph ) -> ( ( [. y / x ]. ph -> [. y / x ]. ps ) <-> ( [. ( iota x ph ) / x ]. ph -> [. ( iota x ph ) / x ]. ps ) ) )
13	9, 12	syl5ib 219	. . . . . 6 |- ( y = ( iota x ph ) -> ( A. x ( ph -> ps ) -> ( [. ( iota x ph ) / x ]. ph -> [. ( iota x ph ) / x ]. ps ) ) )
14	13	com23 78	. . . . 5 |- ( y = ( iota x ph ) -> ( [. ( iota x ph ) / x ]. ph -> ( A. x ( ph -> ps ) -> [. ( iota x ph ) / x ]. ps ) ) )
15	5, 14	syl 16	. . . 4 |- ( A. x ( ph <-> x = y ) -> ( [. ( iota x ph ) / x ]. ph -> ( A. x ( ph -> ps ) -> [. ( iota x ph ) / x ]. ps ) ) )
16	15	exlimiv 1688	. . 3 |- ( E. y A. x ( ph <-> x = y ) -> ( [. ( iota x ph ) / x ]. ph -> ( A. x ( ph -> ps ) -> [. ( iota x ph ) / x ]. ps ) ) )
17	2, 3, 16	sylc 60	. 2 |- ( E! x ph -> ( A. x ( ph -> ps ) -> [. ( iota x ph ) / x ]. ps ) )
18		iotaexeu 29670	. . . . 5 |- ( E! x ph -> ( iota x ph ) e. _V )
19	10, 11	anbi12d 710	. . . . . . . 8 |- ( y = ( iota x ph ) -> ( ( [. y / x ]. ph /\ [. y / x ]. ps ) <-> ( [. ( iota x ph ) / x ]. ph /\ [. ( iota x ph ) / x ]. ps ) ) )
20	19	imbi1d 317	. . . . . . 7 |- ( y = ( iota x ph ) -> ( ( ( [. y / x ]. ph /\ [. y / x ]. ps ) -> E. x ( ph /\ ps ) ) <-> ( ( [. ( iota x ph ) / x ]. ph /\ [. ( iota x ph ) / x ]. ps ) -> E. x ( ph /\ ps ) ) ) )
21		sbcan 3228	. . . . . . . 8 |- ( [. y / x ]. ( ph /\ ps ) <-> ( [. y / x ]. ph /\ [. y / x ]. ps ) )
22		spesbc 3278	. . . . . . . 8 |- ( [. y / x ]. ( ph /\ ps ) -> E. x ( ph /\ ps ) )
23	21, 22	sylbir 213	. . . . . . 7 |- ( ( [. y / x ]. ph /\ [. y / x ]. ps ) -> E. x ( ph /\ ps ) )
24	20, 23	vtoclg 3029	. . . . . 6 |- ( ( iota x ph ) e. _V -> ( ( [. ( iota x ph ) / x ]. ph /\ [. ( iota x ph ) / x ]. ps ) -> E. x ( ph /\ ps ) ) )
25	24	expd 436	. . . . 5 |- ( ( iota x ph ) e. _V -> ( [. ( iota x ph ) / x ]. ph -> ( [. ( iota x ph ) / x ]. ps -> E. x ( ph /\ ps ) ) ) )
26	18, 3, 25	sylc 60	. . . 4 |- ( E! x ph -> ( [. ( iota x ph ) / x ]. ps -> E. x ( ph /\ ps ) ) )
27	26	anc2li 557	. . 3 |- ( E! x ph -> ( [. ( iota x ph ) / x ]. ps -> ( E! x ph /\ E. x ( ph /\ ps ) ) ) )
28		eupicka 2343	. . 3 |- ( ( E! x ph /\ E. x ( ph /\ ps ) ) -> A. x ( ph -> ps ) )
29	27, 28	syl6 33	. 2 |- ( E! x ph -> ( [. ( iota x ph ) / x ]. ps -> A. x ( ph -> ps ) ) )
30	17, 29	impbid 191	1 |- ( E! x ph -> ( A. x ( ph -> ps ) <-> [. ( iota x ph ) / x ]. ps ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   A.wal 1367   = wceq 1369   E.wex 1586   [wsb 1700   e. wcel 1756   E!weu 2253   _Vcvv 2971   [.wsbc 3185   iotacio 5378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ral 2719  df-rex 2720  df-v 2973  df-sbc 3186  df-un 3332  df-sn 3877  df-pr 3879  df-uni 4091  df-iota 5380

This theorem is referenced by:  sbaniota  29687