Theorem iotavalb 36826

Description: Theorem *14.202 in [WhiteheadRussell] p. 189. A biconditional version of iotaval 5580. (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 5580	. 2 |- ( A. x ( ph <-> x = y ) -> ( iota x ph ) = y )
2		iotasbc 36815	. . . 4 |- ( E! x ph -> ( [. ( iota x ph ) / z ]. z = y <-> E. z ( A. x ( ph <-> x = z ) /\ z = y ) ) )
3		iotaexeu 36814	. . . . 5 |- ( E! x ph -> ( iota x ph ) e. _V )
4		eqsbc3 3319	. . . . 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 263	. . 3 |- ( E! x ph -> ( E. z ( A. x ( ph <-> x = z ) /\ z = y ) <-> ( iota x ph ) = y ) )
7		equequ2 1879	. . . . . . 7 |- ( z = y -> ( x = z <-> x = y ) )
8	7	bibi2d 324	. . . . . 6 |- ( z = y -> ( ( ph <-> x = z ) <-> ( ph <-> x = y ) ) )
9	8	albidv 1778	. . . . 5 |- ( z = y -> ( A. x ( ph <-> x = z ) <-> A. x ( ph <-> x = y ) ) )
10	9	biimpac 493	. . . 4 |- ( ( A. x ( ph <-> x = z ) /\ z = y ) -> A. x ( ph <-> x = y ) )
11	10	exlimiv 1787	. . 3 |- ( E. z ( A. x ( ph <-> x = z ) /\ z = y ) -> A. x ( ph <-> x = y ) )
12	6, 11	syl6bir 237	. 2 |- ( E! x ph -> ( ( iota x ph ) = y -> A. x ( ph <-> x = y ) ) )
13	1, 12	impbid2 209	1 |- ( E! x ph -> ( A. x ( ph <-> x = y ) <-> ( iota x ph ) = y ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   A.wal 1453   = wceq 1455   E.wex 1674   e. wcel 1898   E!weu 2310   _Vcvv 3057   [.wsbc 3279   iotacio 5567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-rex 2755  df-v 3059  df-sbc 3280  df-un 3421  df-sn 3981  df-pr 3983  df-uni 4213  df-iota 5569

This theorem is referenced by:  iotavalsb  36829