Theorem iotan0 6528

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 3036	. . . . . 6 ⊢ ((𝐴 = (℩𝑥𝜑) ∧ 𝐴 ≠ ∅) → (℩𝑥𝜑) ≠ ∅)
2	1	expcom 419	. . . . 5 ⊢ (𝐴 ≠ ∅ → (𝐴 = (℩𝑥𝜑) → (℩𝑥𝜑) ≠ ∅))
3		iotanul 6518	. . . . . 6 ⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
4	3	necon1ai 2982	. . . . 5 ⊢ ((℩𝑥𝜑) ≠ ∅ → ∃!𝑥𝜑)
5	2, 4	syl6 36	. . . 4 ⊢ (𝐴 ≠ ∅ → (𝐴 = (℩𝑥𝜑) → ∃!𝑥𝜑))
6	5	a1i 11	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ≠ ∅ → (𝐴 = (℩𝑥𝜑) → ∃!𝑥𝜑)))
7	6	3imp 1128	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = (℩𝑥𝜑)) → ∃!𝑥𝜑)
8		eqcom 2767	. . . . 5 ⊢ (𝐴 = (℩𝑥𝜑) ↔ (℩𝑥𝜑) = 𝐴)
9		iotan0.1	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
10	9	iota2 6527	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴))
11	10	biimprd 251	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃!𝑥𝜑) → ((℩𝑥𝜑) = 𝐴 → 𝜓))
12	8, 11	biimtrid 245	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃!𝑥𝜑) → (𝐴 = (℩𝑥𝜑) → 𝜓))
13	12	impancom 457	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = (℩𝑥𝜑)) → (∃!𝑥𝜑 → 𝜓))
14	13	3adant2 1149	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = (℩𝑥𝜑)) → (∃!𝑥𝜑 → 𝜓))
15	7, 14	mpd 16	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = (℩𝑥𝜑)) → 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∃!weu 2593   ≠ wne 2955  ∅c0 4278  ℩cio 6492

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

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-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-sn 4585  df-pr 4587  df-uni 4867  df-iota 6494

This theorem is used by:  sgrpidmnd  18933