Theorem iotavalb 36149

Description: Theorem *14.202 in [WhiteheadRussell] p. 189. A biconditional version of iotaval 5498. (Contributed by Andrew Salmon, 11-Jul-2011.)

Assertion

Ref	Expression
iotavalb	|- ( E! x ph -> ( A. x ( ph <-> x = y ) <-> ( iota x ph ) = y ) )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem iotavalb

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iotaval 5498	. 2 |- ( A. x ( ph <-> x = y ) -> ( iota x ph ) = y )
2		iotasbc 36138	. . . 4 |- ( E! x ph -> ( [. ( iota x ph ) / z ]. z = y <-> E. z ( A. x ( ph <-> x = z ) /\ z = y ) ) )
3		iotaexeu 36137	. . . . 5 |- ( E! x ph -> ( iota x ph ) e. _V )
4		eqsbc3 3314	. . . . 5 |- ( ( iota x ph ) e. _V -> ( [. ( iota x ph ) / z ]. z = y <-> ( iota x ph ) = y ) )
5	3, 4	syl 17	. . . 4 |- ( E! x ph -> ( [. ( iota x ph ) / z ]. z = y <-> ( iota x ph ) = y ) )
6	2, 5	bitr3d 255	. . 3 |- ( E! x ph -> ( E. z ( A. x ( ph <-> x = z ) /\ z = y ) <-> ( iota x ph ) = y ) )
7		equequ2 1821	. . . . . . 7 |- ( z = y -> ( x = z <-> x = y ) )
8	7	bibi2d 316	. . . . . 6 |- ( z = y -> ( ( ph <-> x = z ) <-> ( ph <-> x = y ) ) )
9	8	albidv 1732	. . . . 5 |- ( z = y -> ( A. x ( ph <-> x = z ) <-> A. x ( ph <-> x = y ) ) )
10	9	biimpac 484	. . . 4 |- ( ( A. x ( ph <-> x = z ) /\ z = y ) -> A. x ( ph <-> x = y ) )
11	10	exlimiv 1741	. . 3 |- ( E. z ( A. x ( ph <-> x = z ) /\ z = y ) -> A. x ( ph <-> x = y ) )
12	6, 11	syl6bir 229	. 2 |- ( E! x ph -> ( ( iota x ph ) = y -> A. x ( ph <-> x = y ) ) )
13	1, 12	impbid2 204	1 |- ( E! x ph -> ( A. x ( ph <-> x = y ) <-> ( iota x ph ) = y ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   A.wal 1401   = wceq 1403   E.wex 1631   e. wcel 1840   E!weu 2236   _Vcvv 3056   [.wsbc 3274   iotacio 5485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-rex 2757  df-v 3058  df-sbc 3275  df-un 3416  df-sn 3970  df-pr 3972  df-uni 4189  df-iota 5487

This theorem is referenced by:  iotavalsb  36152