Theorem iota5 4907

Description: A method for computing iota. (Contributed by NM, 17-Sep-2013.)

Hypothesis

Ref	Expression
iota5.1	|- ( ( ph /\ A e. V ) -> ( ps <-> x = A ) )

Assertion

Ref	Expression
iota5	|- ( ( ph /\ A e. V ) -> ( iota x ps ) = A )

Distinct variable groups:   x, A   x, V   ph, x

Allowed substitution hint:   ps( x)

Proof of Theorem iota5

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iota5.1	. . 3 |- ( ( ph /\ A e. V ) -> ( ps <-> x = A ) )
2	1	alrimiv 1795	. 2 |- ( ( ph /\ A e. V ) -> A. x ( ps <-> x = A ) )
3		eqeq2 2090	. . . . . . 7 |- ( y = A -> ( x = y <-> x = A ) )
4	3	bibi2d 230	. . . . . 6 |- ( y = A -> ( ( ps <-> x = y ) <-> ( ps <-> x = A ) ) )
5	4	albidv 1745	. . . . 5 |- ( y = A -> ( A. x ( ps <-> x = y ) <-> A. x ( ps <-> x = A ) ) )
6		eqeq2 2090	. . . . 5 |- ( y = A -> ( ( iota x ps ) = y <-> ( iota x ps ) = A ) )
7	5, 6	imbi12d 232	. . . 4 |- ( y = A -> ( ( A. x ( ps <-> x = y ) -> ( iota x ps ) = y ) <-> ( A. x ( ps <-> x = A ) -> ( iota x ps ) = A ) ) )
8		iotaval 4898	. . . 4 |- ( A. x ( ps <-> x = y ) -> ( iota x ps ) = y )
9	7, 8	vtoclg 2658	. . 3 |- ( A e. V -> ( A. x ( ps <-> x = A ) -> ( iota x ps ) = A ) )
10	9	adantl 271	. 2 |- ( ( ph /\ A e. V ) -> ( A. x ( ps <-> x = A ) -> ( iota x ps ) = A ) )
11	2, 10	mpd 13	1 |- ( ( ph /\ A e. V ) -> ( iota x ps ) = A )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1282   = wceq 1284   e. wcel 1433   iotacio 4885

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-sn 3404  df-pr 3405  df-uni 3602  df-iota 4887

This theorem is referenced by: (None)