Theorem iotavalb 31239

Description: Theorem *14.202 in [WhiteheadRussell] p. 189. A biconditional version of iotaval 5568. (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 5568	. 2 |- ( A. x ( ph <-> x = y ) -> ( iota x ph ) = y )
2		iotasbc 31228	. . . 4 |- ( E! x ph -> ( [. ( iota x ph ) / z ]. z = y <-> E. z ( A. x ( ph <-> x = z ) /\ z = y ) ) )
3		iotaexeu 31227	. . . . 5 |- ( E! x ph -> ( iota x ph ) e. _V )
4		eqsbc3 3376	. . . . 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 1748	. . . . . . 7 |- ( z = y -> ( x = z <-> x = y ) )
8	7	bibi2d 318	. . . . . 6 |- ( z = y -> ( ( ph <-> x = z ) <-> ( ph <-> x = y ) ) )
9	8	albidv 1689	. . . . 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 1698	. . 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 1377   = wceq 1379   E.wex 1596   e. wcel 1767   E!weu 2275   _Vcvv 3118   [.wsbc 3336   iotacio 5555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rex 2823  df-v 3120  df-sbc 3337  df-un 3486  df-sn 4034  df-pr 4036  df-uni 4252  df-iota 5557

This theorem is referenced by:  iotavalsb  31242