Theorem iotam 5309

Description: Representation of "the unique element such that ph " with a class expression A which is inhabited (that means that "the unique element such that ph " exists). (Contributed by AV, 30-Jan-2024.)

Hypothesis

Ref	Expression
iotam.1	|- ( x = A -> ( ph <-> ps ) )

Assertion

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

Distinct variable groups:   x, A   w, A   ps, x

Allowed substitution hints:   ph( x, w)   ps( w)   V( x, w)

Proof of Theorem iotam

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1w 2290	. . . . 5 |- ( w = z -> ( w e. A <-> z e. A ) )
2	1	cbvexv 1965	. . . 4 |- ( E. w w e. A <-> E. z z e. A )
3		simprr 531	. . . . . . . 8 |- ( ( z e. A /\ ( A e. V /\ A = ( iota x ph ) ) ) -> A = ( iota x ph ) )
4	3	eqcomd 2235	. . . . . . 7 |- ( ( z e. A /\ ( A e. V /\ A = ( iota x ph ) ) ) -> ( iota x ph ) = A )
5		simprl 529	. . . . . . . 8 |- ( ( z e. A /\ ( A e. V /\ A = ( iota x ph ) ) ) -> A e. V )
6		simpl 109	. . . . . . . . . 10 |- ( ( z e. A /\ ( A e. V /\ A = ( iota x ph ) ) ) -> z e. A )
7	6, 3	eleqtrd 2308	. . . . . . . . 9 |- ( ( z e. A /\ ( A e. V /\ A = ( iota x ph ) ) ) -> z e. ( iota x ph ) )
8		eliotaeu 5306	. . . . . . . . 9 |- ( z e. ( iota x ph ) -> E! x ph )
9	7, 8	syl 14	. . . . . . . 8 |- ( ( z e. A /\ ( A e. V /\ A = ( iota x ph ) ) ) -> E! x ph )
10		iotam.1	. . . . . . . . 9 |- ( x = A -> ( ph <-> ps ) )
11	10	iota2 5307	. . . . . . . 8 |- ( ( A e. V /\ E! x ph ) -> ( ps <-> ( iota x ph ) = A ) )
12	5, 9, 11	syl2anc 411	. . . . . . 7 |- ( ( z e. A /\ ( A e. V /\ A = ( iota x ph ) ) ) -> ( ps <-> ( iota x ph ) = A ) )
13	4, 12	mpbird 167	. . . . . 6 |- ( ( z e. A /\ ( A e. V /\ A = ( iota x ph ) ) ) -> ps )
14	13	ex 115	. . . . 5 |- ( z e. A -> ( ( A e. V /\ A = ( iota x ph ) ) -> ps ) )
15	14	exlimiv 1644	. . . 4 |- ( E. z z e. A -> ( ( A e. V /\ A = ( iota x ph ) ) -> ps ) )
16	2, 15	sylbi 121	. . 3 |- ( E. w w e. A -> ( ( A e. V /\ A = ( iota x ph ) ) -> ps ) )
17	16	3impib 1225	. 2 |- ( ( E. w w e. A /\ A e. V /\ A = ( iota x ph ) ) -> ps )
18	17	3com12 1231	1 |- ( ( A e. V /\ E. w w e. A /\ A = ( iota x ph ) ) -> ps )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   E.wex 1538   E!weu 2077   e. wcel 2200   iotacio 5275

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-sn 3672  df-pr 3673  df-uni 3888  df-iota 5277

This theorem is referenced by:  sgrpidmndm  13448