Theorem iotam 5272

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 2267	. . . . 5 |- ( w = z -> ( w e. A <-> z e. A ) )
2	1	cbvexv 1943	. . . 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 2212	. . . . . . 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 2285	. . . . . . . . 9 |- ( ( z e. A /\ ( A e. V /\ A = ( iota x ph ) ) ) -> z e. ( iota x ph ) )
8		eliotaeu 5269	. . . . . . . . 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 5270	. . . . . . . 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 1622	. . . 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 1204	. 2 |- ( ( E. w w e. A /\ A e. V /\ A = ( iota x ph ) ) -> ps )
18	17	3com12 1210	1 |- ( ( A e. V /\ E. w w e. A /\ A = ( iota x ph ) ) -> ps )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   E.wex 1516   E!weu 2055   e. wcel 2177   iotacio 5239

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-rex 2491  df-v 2775  df-sbc 3003  df-un 3174  df-sn 3644  df-pr 3645  df-uni 3857  df-iota 5241

This theorem is referenced by:  sgrpidmndm  13327