Theorem iotam 5318

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

Hypothesis

Ref	Expression
iotam.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
iotam	⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑤 𝑤 ∈ 𝐴 ∧ 𝐴 = (℩𝑥𝜑)) → 𝜓)

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

Allowed substitution hints:   𝜑(𝑥,𝑤)   𝜓(𝑤)   𝑉(𝑥,𝑤)

Proof of Theorem iotam

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1w 2292	. . . . 5 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
2	1	cbvexv 1967	. . . 4 ⊢ (∃𝑤 𝑤 ∈ 𝐴 ↔ ∃𝑧 𝑧 ∈ 𝐴)
3		simprr 533	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐴 ∧ (𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑))) → 𝐴 = (℩𝑥𝜑))
4	3	eqcomd 2237	. . . . . . 7 ⊢ ((𝑧 ∈ 𝐴 ∧ (𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑))) → (℩𝑥𝜑) = 𝐴)
5		simprl 531	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐴 ∧ (𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑))) → 𝐴 ∈ 𝑉)
6		simpl 109	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝐴 ∧ (𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑))) → 𝑧 ∈ 𝐴)
7	6, 3	eleqtrd 2310	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐴 ∧ (𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑))) → 𝑧 ∈ (℩𝑥𝜑))
8		eliotaeu 5315	. . . . . . . . 9 ⊢ (𝑧 ∈ (℩𝑥𝜑) → ∃!𝑥𝜑)
9	7, 8	syl 14	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐴 ∧ (𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑))) → ∃!𝑥𝜑)
10		iotam.1	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
11	10	iota2 5316	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴))
12	5, 9, 11	syl2anc 411	. . . . . . 7 ⊢ ((𝑧 ∈ 𝐴 ∧ (𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑))) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴))
13	4, 12	mpbird 167	. . . . . 6 ⊢ ((𝑧 ∈ 𝐴 ∧ (𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑))) → 𝜓)
14	13	ex 115	. . . . 5 ⊢ (𝑧 ∈ 𝐴 → ((𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑)) → 𝜓))
15	14	exlimiv 1646	. . . 4 ⊢ (∃𝑧 𝑧 ∈ 𝐴 → ((𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑)) → 𝜓))
16	2, 15	sylbi 121	. . 3 ⊢ (∃𝑤 𝑤 ∈ 𝐴 → ((𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑)) → 𝜓))
17	16	3impib 1227	. 2 ⊢ ((∃𝑤 𝑤 ∈ 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑)) → 𝜓)
18	17	3com12 1233	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑤 𝑤 ∈ 𝐴 ∧ 𝐴 = (℩𝑥𝜑)) → 𝜓)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1004   = wceq 1397  ∃wex 1540  ∃!weu 2079   ∈ wcel 2202  ℩cio 5284

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-sn 3675  df-pr 3676  df-uni 3894  df-iota 5286

This theorem is referenced by:  sgrpidmndm  13505