Theorem iotam 5190

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 2231	. . . . 5 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
2	1	cbvexv 1911	. . . 4 ⊢ (∃𝑤 𝑤 ∈ 𝐴 ↔ ∃𝑧 𝑧 ∈ 𝐴)
3		simprr 527	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐴 ∧ (𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑))) → 𝐴 = (℩𝑥𝜑))
4	3	eqcomd 2176	. . . . . . 7 ⊢ ((𝑧 ∈ 𝐴 ∧ (𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑))) → (℩𝑥𝜑) = 𝐴)
5		simprl 526	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐴 ∧ (𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑))) → 𝐴 ∈ 𝑉)
6		simpl 108	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝐴 ∧ (𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑))) → 𝑧 ∈ 𝐴)
7	6, 3	eleqtrd 2249	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐴 ∧ (𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑))) → 𝑧 ∈ (℩𝑥𝜑))
8		eliotaeu 5187	. . . . . . . . 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 5188	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴))
12	5, 9, 11	syl2anc 409	. . . . . . 7 ⊢ ((𝑧 ∈ 𝐴 ∧ (𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑))) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴))
13	4, 12	mpbird 166	. . . . . 6 ⊢ ((𝑧 ∈ 𝐴 ∧ (𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑))) → 𝜓)
14	13	ex 114	. . . . 5 ⊢ (𝑧 ∈ 𝐴 → ((𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑)) → 𝜓))
15	14	exlimiv 1591	. . . 4 ⊢ (∃𝑧 𝑧 ∈ 𝐴 → ((𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑)) → 𝜓))
16	2, 15	sylbi 120	. . 3 ⊢ (∃𝑤 𝑤 ∈ 𝐴 → ((𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑)) → 𝜓))
17	16	3impib 1196	. 2 ⊢ ((∃𝑤 𝑤 ∈ 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑)) → 𝜓)
18	17	3com12 1202	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑤 𝑤 ∈ 𝐴 ∧ 𝐴 = (℩𝑥𝜑)) → 𝜓)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 973   = wceq 1348  ∃wex 1485  ∃!weu 2019   ∈ wcel 2141  ℩cio 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