Theorem aiotaexim 48441

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 6555 which requires 𝐴 to be nonempty instead of being a set. That gives a more natural statement, closer to statements like spcgv 3554. (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 48435	. . . 4 ⊢ (℩'𝑥𝜑) = if(∃!𝑥𝜑, ∩ {𝑥 ∣ 𝜑}, V)
2	1	eqeq2i 2776	. . 3 ⊢ (𝐴 = (℩'𝑥𝜑) ↔ 𝐴 = if(∃!𝑥𝜑, ∩ {𝑥 ∣ 𝜑}, V))
3		iftrue 4491	. . . . . 6 ⊢ (∃!𝑥𝜑 → if(∃!𝑥𝜑, ∩ {𝑥 ∣ 𝜑}, V) = ∩ {𝑥 ∣ 𝜑})
4	3	eqeq2d 2774	. . . . 5 ⊢ (∃!𝑥𝜑 → (𝐴 = if(∃!𝑥𝜑, ∩ {𝑥 ∣ 𝜑}, V) ↔ 𝐴 = ∩ {𝑥 ∣ 𝜑}))
5		iotain 6539	. . . . . . 7 ⊢ (∃!𝑥𝜑 → ∩ {𝑥 ∣ 𝜑} = (℩𝑥𝜑))
6	5	eqeq2d 2774	. . . . . 6 ⊢ (∃!𝑥𝜑 → (𝐴 = ∩ {𝑥 ∣ 𝜑} ↔ 𝐴 = (℩𝑥𝜑)))
7		eqcom 2770	. . . . . . 7 ⊢ (𝐴 = (℩𝑥𝜑) ↔ (℩𝑥𝜑) = 𝐴)
8		aiotaexim.1	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
9	8	iota2 6554	. . . . . . . . . 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 4494	. . . . . 6 ⊢ (¬ ∃!𝑥𝜑 → if(∃!𝑥𝜑, ∩ {𝑥 ∣ 𝜑}, V) = V)
17	16	eqeq2d 2774	. . . . 5 ⊢ (¬ ∃!𝑥𝜑 → (𝐴 = if(∃!𝑥𝜑, ∩ {𝑥 ∣ 𝜑}, V) ↔ 𝐴 = V))
18		nvelim 5280	. . . . . 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 1570   ∈ wcel 2147  ∃!weu 2596  {cab 2741  Vcvv 3454  ifcif 4485  ∩ cint 4912  ℩cio 6518  ℩'caiota 48426

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-uni 4873  df-int 4913  df-iota 6520  df-aiota 48428

This theorem is used by: (None)