Theorem iota5 5253

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 1897	. 2 |- ( ( ph /\ A e. V ) -> A. x ( ps <-> x = A ) )
3		eqeq2 2215	. . . . . . 7 |- ( y = A -> ( x = y <-> x = A ) )
4	3	bibi2d 232	. . . . . 6 |- ( y = A -> ( ( ps <-> x = y ) <-> ( ps <-> x = A ) ) )
5	4	albidv 1847	. . . . 5 |- ( y = A -> ( A. x ( ps <-> x = y ) <-> A. x ( ps <-> x = A ) ) )
6		eqeq2 2215	. . . . 5 |- ( y = A -> ( ( iota x ps ) = y <-> ( iota x ps ) = A ) )
7	5, 6	imbi12d 234	. . . 4 |- ( y = A -> ( ( A. x ( ps <-> x = y ) -> ( iota x ps ) = y ) <-> ( A. x ( ps <-> x = A ) -> ( iota x ps ) = A ) ) )
8		iotaval 5243	. . . 4 |- ( A. x ( ps <-> x = y ) -> ( iota x ps ) = y )
9	7, 8	vtoclg 2833	. . 3 |- ( A e. V -> ( A. x ( ps <-> x = A ) -> ( iota x ps ) = A ) )
10	9	adantl 277	. 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 104   <-> wb 105   A.wal 1371   = wceq 1373   e. wcel 2176   iotacio 5230

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-sn 3639  df-pr 3640  df-uni 3851  df-iota 5232

This theorem is referenced by:  fsum3  11698  fprodseq  11894  gsumfzval  13223