Theorem iotaval 5252

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 5242	. 2 |- ( iota x ph ) = U. { z | A. x ( ph <-> x = z ) }
2		vex 2776	. . . . . . 7 |- y e. _V
3		sbeqalb 3059	. . . . . . . 8 |- ( y e. _V -> ( ( A. x ( ph <-> x = y ) /\ A. x ( ph <-> x = z ) ) -> y = z ) )
4		equcomi 1728	. . . . . . . 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 1737	. . . . . . . . . 10 |- ( y = z -> ( x = y <-> x = z ) )
9	8	equcoms 1732	. . . . . . . . 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 1903	. . . . . 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 1898	. . 3 |- ( A. x ( ph <-> x = y ) -> A. z ( A. x ( ph <-> x = z ) <-> z = y ) )
16		uniabio 5251	. . 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 2251	1 |- ( A. x ( ph <-> x = y ) -> ( iota x ph ) = y )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1371   = wceq 1373   e. wcel 2177   {cab 2192   _Vcvv 2773   U.cuni 3856   iotacio 5239

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-rex 2491  df-v 2775  df-sbc 3003  df-un 3174  df-sn 3644  df-pr 3645  df-uni 3857  df-iota 5241

This theorem is referenced by:  iotauni  5253  iota1  5255  euiotaex  5257  iota4  5260  iota5  5262