Theorem iotaval 5344

Description: Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.)

Assertion

Ref	Expression
iotaval	|- ( A. x ( ph <-> x = y ) -> ( iota x ph ) = y )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem iotaval

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfiota2 5333	. 2 |- ( iota x ph ) = U. { z | A. x ( ph <-> x = z ) }
2		vex 2824	. . . . . . 7 |- y e. _V
3		sbeqalb 3108	. . . . . . . 8 |- ( y e. _V -> ( ( A. x ( ph <-> x = y ) /\ A. x ( ph <-> x = z ) ) -> y = z ) )
4		equcomi 1756	. . . . . . . 8 |- ( y = z -> z = y )
5	3, 4	syl6 33	. . . . . . 7 |- ( y e. _V -> ( ( A. x ( ph <-> x = y ) /\ A. x ( ph <-> x = z ) ) -> z = y ) )
6	2, 5	ax-mp 5	. . . . . 6 |- ( ( A. x ( ph <-> x = y ) /\ A. x ( ph <-> x = z ) ) -> z = y )
7	6	ex 115	. . . . 5 |- ( A. x ( ph <-> x = y ) -> ( A. x ( ph <-> x = z ) -> z = y ) )
8		equequ2 1765	. . . . . . . . . 10 |- ( y = z -> ( x = y <-> x = z ) )
9	8	equcoms 1760	. . . . . . . . 9 |- ( z = y -> ( x = y <-> x = z ) )
10	9	bibi2d 232	. . . . . . . 8 |- ( z = y -> ( ( ph <-> x = y ) <-> ( ph <-> x = z ) ) )
11	10	biimpd 144	. . . . . . 7 |- ( z = y -> ( ( ph <-> x = y ) -> ( ph <-> x = z ) ) )
12	11	alimdv 1932	. . . . . 6 |- ( z = y -> ( A. x ( ph <-> x = y ) -> A. x ( ph <-> x = z ) ) )
13	12	com12 30	. . . . 5 |- ( A. x ( ph <-> x = y ) -> ( z = y -> A. x ( ph <-> x = z ) ) )
14	7, 13	impbid 129	. . . 4 |- ( A. x ( ph <-> x = y ) -> ( A. x ( ph <-> x = z ) <-> z = y ) )
15	14	alrimiv 1927	. . 3 |- ( A. x ( ph <-> x = y ) -> A. z ( A. x ( ph <-> x = z ) <-> z = y ) )
16		uniabio 5343	. . 3 |- ( A. z ( A. x ( ph <-> x = z ) <-> z = y ) -> U. { z | A. x ( ph <-> x = z ) } = y )
17	15, 16	syl 14	. 2 |- ( A. x ( ph <-> x = y ) -> U. { z | A. x ( ph <-> x = z ) } = y )
18	1, 17	eqtrid 2283	1 |- ( A. x ( ph <-> x = y ) -> ( iota x ph ) = y )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1400   = wceq 1402   e. wcel 2209   {cab 2224   _Vcvv 2821   U.cuni 3930   iotacio 5330

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-sn 3711  df-pr 3712  df-uni 3931  df-iota 5332

This theorem is referenced by:  iotauni  5345  iota1  5347  euiotaex  5349  iota4  5352  iota5  5354