Theorem iotan0 6550

Description: Representation of "the unique element such that 𝜑 " with a class expression 𝐴 which is not the empty set (that means that "the unique element such that 𝜑 " exists). (Contributed by AV, 30-Jan-2024.) Remove sethood antecedent. (Revised by Eric Schmidt, 17-Oct-2026.)

Hypothesis

Ref	Expression
iotan0.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
iotan0	⊢ ((𝐴 ≠ ∅ ∧ 𝐴 = (℩𝑥𝜑)) → 𝜓)

Distinct variable groups:   𝑥,𝐴   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem iotan0

Step	Hyp	Ref	Expression
1		pm13.18 3040	. . . . 5 ⊢ ((𝐴 = (℩𝑥𝜑) ∧ 𝐴 ≠ ∅) → (℩𝑥𝜑) ≠ ∅)
2	1	expcom 419	. . . 4 ⊢ (𝐴 ≠ ∅ → (𝐴 = (℩𝑥𝜑) → (℩𝑥𝜑) ≠ ∅))
3		iotanul 6540	. . . . 5 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
4	3	necon1ai 2986	. . . 4 ⊢ ((℩𝑥𝜑) ≠ ∅ → ∃!𝑥𝜑)
5	2, 4	syl6 36	. . 3 ⊢ (𝐴 ≠ ∅ → (𝐴 = (℩𝑥𝜑) → ∃!𝑥𝜑))
6	5	imp 412	. 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 = (℩𝑥𝜑)) → ∃!𝑥𝜑)
7		iotaex 6536	. . . . 5 ⊢ (℩𝑥𝜑) ∈ V
8		eleq1 2852	. . . . 5 ⊢ (𝐴 = (℩𝑥𝜑) → (𝐴 ∈ V ↔ (℩𝑥𝜑) ∈ V))
9	7, 8	mpbiri 261	. . . 4 ⊢ (𝐴 = (℩𝑥𝜑) → 𝐴 ∈ V)
10		eqcom 2770	. . . . . 6 ⊢ (𝐴 = (℩𝑥𝜑) ↔ (℩𝑥𝜑) = 𝐴)
11		iotan0.1	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
12	11	iota2 6549	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴))
13	12	biimprd 251	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → ((℩𝑥𝜑) = 𝐴 → 𝜓))
14	10, 13	biimtrid 245	. . . . 5 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → (𝐴 = (℩𝑥𝜑) → 𝜓))
15	14	impancom 457	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐴 = (℩𝑥𝜑)) → (∃!𝑥𝜑 → 𝜓))
16	9, 15	mpancom 701	. . 3 ⊢ (𝐴 = (℩𝑥𝜑) → (∃!𝑥𝜑 → 𝜓))
17	16	adantl 487	. 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 = (℩𝑥𝜑)) → (∃!𝑥𝜑 → 𝜓))
18	6, 17	mpd 16	1 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 = (℩𝑥𝜑)) → 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∃!weu 2596   ≠ wne 2959  Vcvv 3454  ∅c0 4281  ℩cio 6513

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-nul 5264

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-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-sn 4589  df-pr 4591  df-uni 4871  df-iota 6515

This theorem is used by:  sgrpidmnd  18960