Theorem iota5 4887

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 1754	. 2 |- ( ( ph /\ A e. V ) -> A. x ( ps <-> x = A ) )
3		eqeq2 2049	. . . . . . 7 |- ( y = A -> ( x = y <-> x = A ) )
4	3	bibi2d 221	. . . . . 6 |- ( y = A -> ( ( ps <-> x = y ) <-> ( ps <-> x = A ) ) )
5	4	albidv 1705	. . . . 5 |- ( y = A -> ( A. x ( ps <-> x = y ) <-> A. x ( ps <-> x = A ) ) )
6		eqeq2 2049	. . . . 5 |- ( y = A -> ( ( iota x ps ) = y <-> ( iota x ps ) = A ) )
7	5, 6	imbi12d 223	. . . 4 |- ( y = A -> ( ( A. x ( ps <-> x = y ) -> ( iota x ps ) = y ) <-> ( A. x ( ps <-> x = A ) -> ( iota x ps ) = A ) ) )
8		iotaval 4878	. . . 4 |- ( A. x ( ps <-> x = y ) -> ( iota x ps ) = y )
9	7, 8	vtoclg 2613	. . 3 |- ( A e. V -> ( A. x ( ps <-> x = A ) -> ( iota x ps ) = A ) )
10	9	adantl 262	. 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 97   <-> wb 98   A.wal 1241   = wceq 1243   e. wcel 1393   iotacio 4865

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-sn 3381  df-pr 3382  df-uni 3581  df-iota 4867

This theorem is referenced by: (None)