Theorem aiotaexim 48276

Description: Representation of "the unique element such that 𝜑 " with a class expression 𝐴 which is a set (that means that "the unique element such that 𝜑 " exists). Analog to iotan0 6532 which requires 𝐴 to be nonempty instead of being a set. That gives a more natural statement, closer to statements like spcgv 3550. (Proposed by BJ, 18-Oct-2026.) (Contributed by AV, 18-Oct-2026.)

Hypothesis

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

Assertion

Ref	Expression
aiotaexim	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = (℩'𝑥𝜑)) → 𝜓)

Distinct variable groups:   𝑥,𝐴   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem aiotaexim

Step	Hyp	Ref	Expression
1		dfaiota3 48270	. . . 4 ⊢ (℩'𝑥𝜑) = if(∃!𝑥𝜑, ∩ {𝑥 ∣ 𝜑}, V)
2	1	eqeq2i 2773	. . 3 ⊢ (𝐴 = (℩'𝑥𝜑) ↔ 𝐴 = if(∃!𝑥𝜑, ∩ {𝑥 ∣ 𝜑}, V))
3		iftrue 4487	. . . . . 6 ⊢ (∃!𝑥𝜑 → if(∃!𝑥𝜑, ∩ {𝑥 ∣ 𝜑}, V) = ∩ {𝑥 ∣ 𝜑})
4	3	eqeq2d 2771	. . . . 5 ⊢ (∃!𝑥𝜑 → (𝐴 = if(∃!𝑥𝜑, ∩ {𝑥 ∣ 𝜑}, V) ↔ 𝐴 = ∩ {𝑥 ∣ 𝜑}))
5		iotain 6516	. . . . . . 7 ⊢ (∃!𝑥𝜑 → ∩ {𝑥 ∣ 𝜑} = (℩𝑥𝜑))
6	5	eqeq2d 2771	. . . . . 6 ⊢ (∃!𝑥𝜑 → (𝐴 = ∩ {𝑥 ∣ 𝜑} ↔ 𝐴 = (℩𝑥𝜑)))
7		eqcom 2767	. . . . . . 7 ⊢ (𝐴 = (℩𝑥𝜑) ↔ (℩𝑥𝜑) = 𝐴)
8		aiotaexim.1	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
9	8	iota2 6531	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴))
10	9	biimprd 251	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃!𝑥𝜑) → ((℩𝑥𝜑) = 𝐴 → 𝜓))
11	10	expcom 419	. . . . . . . 8 ⊢ (∃!𝑥𝜑 → (𝐴 ∈ 𝑉 → ((℩𝑥𝜑) = 𝐴 → 𝜓)))
12	11	com23 87	. . . . . . 7 ⊢ (∃!𝑥𝜑 → ((℩𝑥𝜑) = 𝐴 → (𝐴 ∈ 𝑉 → 𝜓)))
13	7, 12	biimtrid 245	. . . . . 6 ⊢ (∃!𝑥𝜑 → (𝐴 = (℩𝑥𝜑) → (𝐴 ∈ 𝑉 → 𝜓)))
14	6, 13	sylbid 243	. . . . 5 ⊢ (∃!𝑥𝜑 → (𝐴 = ∩ {𝑥 ∣ 𝜑} → (𝐴 ∈ 𝑉 → 𝜓)))
15	4, 14	sylbid 243	. . . 4 ⊢ (∃!𝑥𝜑 → (𝐴 = if(∃!𝑥𝜑, ∩ {𝑥 ∣ 𝜑}, V) → (𝐴 ∈ 𝑉 → 𝜓)))
16		iffalse 4490	. . . . . 6 ⊢ (¬ ∃!𝑥𝜑 → if(∃!𝑥𝜑, ∩ {𝑥 ∣ 𝜑}, V) = V)
17	16	eqeq2d 2771	. . . . 5 ⊢ (¬ ∃!𝑥𝜑 → (𝐴 = if(∃!𝑥𝜑, ∩ {𝑥 ∣ 𝜑}, V) ↔ 𝐴 = V))
18		nvelim 5273	. . . . . 6 ⊢ (𝐴 = V → ¬ 𝐴 ∈ 𝑉)
19	18	pm2.21d 122	. . . . 5 ⊢ (𝐴 = V → (𝐴 ∈ 𝑉 → 𝜓))
20	17, 19	biimtrdi 256	. . . 4 ⊢ (¬ ∃!𝑥𝜑 → (𝐴 = if(∃!𝑥𝜑, ∩ {𝑥 ∣ 𝜑}, V) → (𝐴 ∈ 𝑉 → 𝜓)))
21	15, 20	pm2.61i 184	. . 3 ⊢ (𝐴 = if(∃!𝑥𝜑, ∩ {𝑥 ∣ 𝜑}, V) → (𝐴 ∈ 𝑉 → 𝜓))
22	2, 21	sylbi 220	. 2 ⊢ (𝐴 = (℩'𝑥𝜑) → (𝐴 ∈ 𝑉 → 𝜓))
23	22	impcom 413	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = (℩'𝑥𝜑)) → 𝜓)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∃!weu 2593  {cab 2738  Vcvv 3450  ifcif 4481  ∩ cint 4906  ℩cio 6495  ℩'caiota 48261

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-uni 4867  df-int 4907  df-iota 6497  df-aiota 48263

This theorem is used by: (None)