Theorem iotavalb 36751

Description: Theorem *14.202 in [WhiteheadRussell] p. 189. A biconditional version of iotaval 5576. (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 5576	. 2 |- ( A. x ( ph <-> x = y ) -> ( iota x ph ) = y )
2		iotasbc 36740	. . . 4 |- ( E! x ph -> ( [. ( iota x ph ) / z ]. z = y <-> E. z ( A. x ( ph <-> x = z ) /\ z = y ) ) )
3		iotaexeu 36739	. . . . 5 |- ( E! x ph -> ( iota x ph ) e. _V )
4		eqsbc3 3339	. . . . 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 258	. . 3 |- ( E! x ph -> ( E. z ( A. x ( ph <-> x = z ) /\ z = y ) <-> ( iota x ph ) = y ) )
7		equequ2 1853	. . . . . . 7 |- ( z = y -> ( x = z <-> x = y ) )
8	7	bibi2d 319	. . . . . 6 |- ( z = y -> ( ( ph <-> x = z ) <-> ( ph <-> x = y ) ) )
9	8	albidv 1761	. . . . 5 |- ( z = y -> ( A. x ( ph <-> x = z ) <-> A. x ( ph <-> x = y ) ) )
10	9	biimpac 488	. . . 4 |- ( ( A. x ( ph <-> x = z ) /\ z = y ) -> A. x ( ph <-> x = y ) )
11	10	exlimiv 1770	. . 3 |- ( E. z ( A. x ( ph <-> x = z ) /\ z = y ) -> A. x ( ph <-> x = y ) )
12	6, 11	syl6bir 232	. 2 |- ( E! x ph -> ( ( iota x ph ) = y -> A. x ( ph <-> x = y ) ) )
13	1, 12	impbid2 207	1 |- ( E! x ph -> ( A. x ( ph <-> x = y ) <-> ( iota x ph ) = y ) )

Syntax hints:   -> wi 4   <-> wb 187   /\ wa 370   A.wal 1435   = wceq 1437   E.wex 1657   e. wcel 1872   E!weu 2269   _Vcvv 3080   [.wsbc 3299   iotacio 5563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-rex 2777  df-v 3082  df-sbc 3300  df-un 3441  df-sn 3999  df-pr 4001  df-uni 4220  df-iota 5565

This theorem is referenced by:  iotavalsb  36754