Theorem iotasbc 38620

Description: Definition *14.01 in [WhiteheadRussell] p. 184. In Principia Mathematica, Russell and Whitehead define iota in terms of a function of ( iota x ph ). Their definition differs in that a function of ( iota x ph ) evaluates to "false" when there isn't a single x that satisfies ph. (Contributed by Andrew Salmon, 11-Jul-2011.)

Assertion

Ref	Expression
iotasbc	|- ( E! x ph -> ( [. ( iota x ph ) / y ]. ps <-> E. y ( A. x ( ph <-> x = y ) /\ ps ) ) )

Distinct variable groups:   x, y   ph, y

Allowed substitution hints:   ph( x)   ps( x, y)

Proof of Theorem iotasbc

Step	Hyp	Ref	Expression
1		sbc5 3460	. 2 |- ( [. ( iota x ph ) / y ]. ps <-> E. y ( y = ( iota x ph ) /\ ps ) )
2		iotaexeu 38619	. . . . . . 7 |- ( E! x ph -> ( iota x ph ) e. _V )
3		eueq 3378	. . . . . . 7 |- ( ( iota x ph ) e. _V <-> E! y y = ( iota x ph ) )
4	2, 3	sylib 208	. . . . . 6 |- ( E! x ph -> E! y y = ( iota x ph ) )
5		df-eu 2474	. . . . . . 7 |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )
6		iotaval 5862	. . . . . . . . . 10 |- ( A. x ( ph <-> x = y ) -> ( iota x ph ) = y )
7	6	eqcomd 2628	. . . . . . . . 9 |- ( A. x ( ph <-> x = y ) -> y = ( iota x ph ) )
8	7	ancri 575	. . . . . . . 8 |- ( A. x ( ph <-> x = y ) -> ( y = ( iota x ph ) /\ A. x ( ph <-> x = y ) ) )
9	8	eximi 1762	. . . . . . 7 |- ( E. y A. x ( ph <-> x = y ) -> E. y ( y = ( iota x ph ) /\ A. x ( ph <-> x = y ) ) )
10	5, 9	sylbi 207	. . . . . 6 |- ( E! x ph -> E. y ( y = ( iota x ph ) /\ A. x ( ph <-> x = y ) ) )
11		eupick 2536	. . . . . 6 |- ( ( E! y y = ( iota x ph ) /\ E. y ( y = ( iota x ph ) /\ A. x ( ph <-> x = y ) ) ) -> ( y = ( iota x ph ) -> A. x ( ph <-> x = y ) ) )
12	4, 10, 11	syl2anc 693	. . . . 5 |- ( E! x ph -> ( y = ( iota x ph ) -> A. x ( ph <-> x = y ) ) )
13	12, 7	impbid1 215	. . . 4 |- ( E! x ph -> ( y = ( iota x ph ) <-> A. x ( ph <-> x = y ) ) )
14	13	anbi1d 741	. . 3 |- ( E! x ph -> ( ( y = ( iota x ph ) /\ ps ) <-> ( A. x ( ph <-> x = y ) /\ ps ) ) )
15	14	exbidv 1850	. 2 |- ( E! x ph -> ( E. y ( y = ( iota x ph ) /\ ps ) <-> E. y ( A. x ( ph <-> x = y ) /\ ps ) ) )
16	1, 15	syl5bb 272	1 |- ( E! x ph -> ( [. ( iota x ph ) / y ]. ps <-> E. y ( A. x ( ph <-> x = y ) /\ ps ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   E!weu 2470   _Vcvv 3200   [.wsbc 3435   iotacio 5849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-v 3202  df-sbc 3436  df-un 3579  df-sn 4178  df-pr 4180  df-uni 4437  df-iota 5851

This theorem is referenced by:  iotasbc2  38621  iotavalb  38631  fvsb  38656