Theorem iotan0 6507

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.)

Hypothesis

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

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝜓,𝑥

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

Proof of Theorem iotan0

Step	Hyp	Ref	Expression
1		pm13.18 3037	. . . . . 6 ⊢ ((𝐴 = (℩𝑥𝜑) ∧ 𝐴 ≠ ∅) → (℩𝑥𝜑) ≠ ∅)
2	1	expcom 417	. . . . 5 ⊢ (𝐴 ≠ ∅ → (𝐴 = (℩𝑥𝜑) → (℩𝑥𝜑) ≠ ∅))
3		iotanul 6497	. . . . . 6 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
4	3	necon1ai 2983	. . . . 5 ⊢ ((℩𝑥𝜑) ≠ ∅ → ∃!𝑥𝜑)
5	2, 4	syl6 35	. . . 4 ⊢ (𝐴 ≠ ∅ → (𝐴 = (℩𝑥𝜑) → ∃!𝑥𝜑))
6	5	a1i 11	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ≠ ∅ → (𝐴 = (℩𝑥𝜑) → ∃!𝑥𝜑)))
7	6	3imp 1122	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = (℩𝑥𝜑)) → ∃!𝑥𝜑)
8		eqcom 2768	. . . . 5 ⊢ (𝐴 = (℩𝑥𝜑) ↔ (℩𝑥𝜑) = 𝐴)
9		iotan0.1	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
10	9	iota2 6506	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴))
11	10	biimprd 250	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃!𝑥𝜑) → ((℩𝑥𝜑) = 𝐴 → 𝜓))
12	8, 11	biimtrid 244	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃!𝑥𝜑) → (𝐴 = (℩𝑥𝜑) → 𝜓))
13	12	impancom 455	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑)) → (∃!𝑥𝜑 → 𝜓))
14	13	3adant2 1143	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = (℩𝑥𝜑)) → (∃!𝑥𝜑 → 𝜓))
15	7, 14	mpd 15	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = (℩𝑥𝜑)) → 𝜓)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∃!weu 2594   ≠ wne 2956  ∅c0 4285  ℩cio 6471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-v 3455  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-sn 4582  df-pr 4584  df-uni 4865  df-iota 6473

This theorem is referenced by:  sgrpidmnd  18756