Theorem iota2df 6530

Description: A condition that allows to represent "the unique element such that 𝜑 " with a class expression 𝐴. (Contributed by NM, 30-Dec-2014.)

Hypotheses

Ref	Expression
iota2df.1	⊢ (𝜑 → 𝐵 ∈ 𝑉)
iota2df.2	⊢ (𝜑 → ∃!𝑥𝜓)
iota2df.3	⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒))
iota2df.4	⊢ Ⅎ𝑥𝜑
iota2df.5	⊢ (𝜑 → Ⅎ𝑥𝜒)
iota2df.6	⊢ (𝜑 → Ⅎ𝑥𝐵)

Assertion

Ref	Expression
iota2df	⊢ (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵))

Proof of Theorem iota2df

Step	Hyp	Ref	Expression
1		iota2df.1	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
2		iota2df.3	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒))
3		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵)
4	3	eqeq2d 2742	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → ((℩𝑥𝜓) = 𝑥 ↔ (℩𝑥𝜓) = 𝐵))
5	2, 4	bibi12d 345	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → ((𝜓 ↔ (℩𝑥𝜓) = 𝑥) ↔ (𝜒 ↔ (℩𝑥𝜓) = 𝐵)))
6		iota2df.2	. . 3 ⊢ (𝜑 → ∃!𝑥𝜓)
7		iota1 6520	. . 3 ⊢ (∃!𝑥𝜓 → (𝜓 ↔ (℩𝑥𝜓) = 𝑥))
8	6, 7	syl 17	. 2 ⊢ (𝜑 → (𝜓 ↔ (℩𝑥𝜓) = 𝑥))
9		iota2df.4	. 2 ⊢ Ⅎ𝑥𝜑
10		iota2df.6	. 2 ⊢ (𝜑 → Ⅎ𝑥𝐵)
11		iota2df.5	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝜒)
12		nfiota1 6497	. . . . 5 ⊢ Ⅎ𝑥(℩𝑥𝜓)
13	12	a1i 11	. . . 4 ⊢ (𝜑 → Ⅎ𝑥(℩𝑥𝜓))
14	13, 10	nfeqd 2912	. . 3 ⊢ (𝜑 → Ⅎ𝑥(℩𝑥𝜓) = 𝐵)
15	11, 14	nfbid 1904	. 2 ⊢ (𝜑 → Ⅎ𝑥(𝜒 ↔ (℩𝑥𝜓) = 𝐵))
16	1, 5, 8, 9, 10, 15	vtocldf 3546	1 ⊢ (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540  Ⅎwnf 1784   ∈ wcel 2105  ∃!weu 2561  Ⅎwnfc 2882  ℩cio 6493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-v 3475  df-un 3953  df-in 3955  df-ss 3965  df-sn 4629  df-pr 4631  df-uni 4909  df-iota 6495

This theorem is referenced by:  iota2d  6531  iota2  6532  riota2df  7392  opiota  8049