Theorem iotam 5190

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 2231	. . . . 5 |- ( w = z -> ( w e. A <-> z e. A ) )
2	1	cbvexv 1911	. . . 4 |- ( E. w w e. A <-> E. z z e. A )
3		simprr 527	. . . . . . . 8 |- ( ( z e. A /\ ( A e. V /\ A = ( iota x ph ) ) ) -> A = ( iota x ph ) )
4	3	eqcomd 2176	. . . . . . 7 |- ( ( z e. A /\ ( A e. V /\ A = ( iota x ph ) ) ) -> ( iota x ph ) = A )
5		simprl 526	. . . . . . . 8 |- ( ( z e. A /\ ( A e. V /\ A = ( iota x ph ) ) ) -> A e. V )
6		simpl 108	. . . . . . . . . 10 |- ( ( z e. A /\ ( A e. V /\ A = ( iota x ph ) ) ) -> z e. A )
7	6, 3	eleqtrd 2249	. . . . . . . . 9 |- ( ( z e. A /\ ( A e. V /\ A = ( iota x ph ) ) ) -> z e. ( iota x ph ) )
8		eliotaeu 5187	. . . . . . . . 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 5188	. . . . . . . 8 |- ( ( A e. V /\ E! x ph ) -> ( ps <-> ( iota x ph ) = A ) )
12	5, 9, 11	syl2anc 409	. . . . . . 7 |- ( ( z e. A /\ ( A e. V /\ A = ( iota x ph ) ) ) -> ( ps <-> ( iota x ph ) = A ) )
13	4, 12	mpbird 166	. . . . . 6 |- ( ( z e. A /\ ( A e. V /\ A = ( iota x ph ) ) ) -> ps )
14	13	ex 114	. . . . 5 |- ( z e. A -> ( ( A e. V /\ A = ( iota x ph ) ) -> ps ) )
15	14	exlimiv 1591	. . . 4 |- ( E. z z e. A -> ( ( A e. V /\ A = ( iota x ph ) ) -> ps ) )
16	2, 15	sylbi 120	. . 3 |- ( E. w w e. A -> ( ( A e. V /\ A = ( iota x ph ) ) -> ps ) )
17	16	3impib 1196	. 2 |- ( ( E. w w e. A /\ A e. V /\ A = ( iota x ph ) ) -> ps )
18	17	3com12 1202	1 |- ( ( A e. V /\ E. w w e. A /\ A = ( iota x ph ) ) -> ps )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 973   = wceq 1348   E.wex 1485   E!weu 2019   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-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  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:  sgrpidmndm  12656