Theorem iota5 5180

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 1867	. 2 |- ( ( ph /\ A e. V ) -> A. x ( ps <-> x = A ) )
3		eqeq2 2180	. . . . . . 7 |- ( y = A -> ( x = y <-> x = A ) )
4	3	bibi2d 231	. . . . . 6 |- ( y = A -> ( ( ps <-> x = y ) <-> ( ps <-> x = A ) ) )
5	4	albidv 1817	. . . . 5 |- ( y = A -> ( A. x ( ps <-> x = y ) <-> A. x ( ps <-> x = A ) ) )
6		eqeq2 2180	. . . . 5 |- ( y = A -> ( ( iota x ps ) = y <-> ( iota x ps ) = A ) )
7	5, 6	imbi12d 233	. . . 4 |- ( y = A -> ( ( A. x ( ps <-> x = y ) -> ( iota x ps ) = y ) <-> ( A. x ( ps <-> x = A ) -> ( iota x ps ) = A ) ) )
8		iotaval 5171	. . . 4 |- ( A. x ( ps <-> x = y ) -> ( iota x ps ) = y )
9	7, 8	vtoclg 2790	. . 3 |- ( A e. V -> ( A. x ( ps <-> x = A ) -> ( iota x ps ) = A ) )
10	9	adantl 275	. 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 103   <-> wb 104   A.wal 1346   = wceq 1348   e. wcel 2141   iotacio 5158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-sn 3589  df-pr 3590  df-uni 3797  df-iota 5160

This theorem is referenced by:  fsum3  11350  fprodseq  11546