Theorem iotam 5343

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 2293	. . . . 5 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
2	1	cbvexv 1968	. . . 4 ⊢ (∃𝑤 𝑤 ∈ 𝐴 ↔ ∃𝑧 𝑧 ∈ 𝐴)
3		simprr 533	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐴 ∧ (𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑))) → 𝐴 = (℩𝑥𝜑))
4	3	eqcomd 2238	. . . . . . 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 2311	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐴 ∧ (𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑))) → 𝑧 ∈ (℩𝑥𝜑))
8		eliotaeu 5340	. . . . . . . . 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 5341	. . . . . . . 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 1647	. . . 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 1228	. 2 ⊢ ((∃𝑤 𝑤 ∈ 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑)) → 𝜓)
18	17	3com12 1234	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑤 𝑤 ∈ 𝐴 ∧ 𝐴 = (℩𝑥𝜑)) → 𝜓)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1005   = wceq 1398  ∃wex 1541  ∃!weu 2080   ∈ wcel 2203  ℩cio 5309

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-v 2814  df-sbc 3042  df-un 3214  df-sn 3694  df-pr 3695  df-uni 3914  df-iota 5311

This theorem is referenced by:  sgrpidmndm  13622