Theorem iotavalb 29689

Description: Theorem *14.202 in [WhiteheadRussell] p. 189. A biconditional version of iotaval 5397. (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 5397	. 2 |- ( A. x ( ph <-> x = y ) -> ( iota x ph ) = y )
2		iotasbc 29678	. . . 4 |- ( E! x ph -> ( [. ( iota x ph ) / z ]. z = y <-> E. z ( A. x ( ph <-> x = z ) /\ z = y ) ) )
3		iotaexeu 29677	. . . . 5 |- ( E! x ph -> ( iota x ph ) e. _V )
4		eqsbc3 3231	. . . . 5 |- ( ( iota x ph ) e. _V -> ( [. ( iota x ph ) / z ]. z = y <-> ( iota x ph ) = y ) )
5	3, 4	syl 16	. . . 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 1737	. . . . . . 7 |- ( z = y -> ( x = z <-> x = y ) )
8	7	bibi2d 318	. . . . . 6 |- ( z = y -> ( ( ph <-> x = z ) <-> ( ph <-> x = y ) ) )
9	8	albidv 1679	. . . . 5 |- ( z = y -> ( A. x ( ph <-> x = z ) <-> A. x ( ph <-> x = y ) ) )
10	9	biimpac 486	. . . 4 |- ( ( A. x ( ph <-> x = z ) /\ z = y ) -> A. x ( ph <-> x = y ) )
11	10	exlimiv 1688	. . 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 369   A.wal 1367   = wceq 1369   E.wex 1586   e. wcel 1756   E!weu 2253   _Vcvv 2977   [.wsbc 3191   iotacio 5384

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 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-rex 2726  df-v 2979  df-sbc 3192  df-un 3338  df-sn 3883  df-pr 3885  df-uni 4097  df-iota 5386

This theorem is referenced by:  iotavalsb  29692