Theorem iotam 5200

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 2236	. . . . 5 |- ( w = z -> ( w e. A <-> z e. A ) )
2	1	cbvexv 1916	. . . 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 2181	. . . . . . 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 2254	. . . . . . . . 9 |- ( ( z e. A /\ ( A e. V /\ A = ( iota x ph ) ) ) -> z e. ( iota x ph ) )
8		eliotaeu 5197	. . . . . . . . 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 5198	. . . . . . . 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 1596	. . . 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 1201	. 2 |- ( ( E. w w e. A /\ A e. V /\ A = ( iota x ph ) ) -> ps )
18	17	3com12 1207	1 |- ( ( A e. V /\ E. w w e. A /\ A = ( iota x ph ) ) -> ps )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 978   = wceq 1353   E.wex 1490   E!weu 2024   e. wcel 2146   iotacio 5168

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-rex 2459  df-v 2737  df-sbc 2961  df-un 3131  df-sn 3595  df-pr 3596  df-uni 3806  df-iota 5170

This theorem is referenced by:  sgrpidmndm  12685