Theorem iota2df 5792

Description: A condition that allows us 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 476	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵)
4	3	eqeq2d 2620	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → ((℩𝑥𝜓) = 𝑥 ↔ (℩𝑥𝜓) = 𝐵))
5	2, 4	bibi12d 334	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → ((𝜓 ↔ (℩𝑥𝜓) = 𝑥) ↔ (𝜒 ↔ (℩𝑥𝜓) = 𝐵)))
6		iota2df.2	. . 3 ⊢ (𝜑 → ∃!𝑥𝜓)
7		iota1 5782	. . 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 5770	. . . . 5 ⊢ Ⅎ𝑥(℩𝑥𝜓)
13	12	a1i 11	. . . 4 ⊢ (𝜑 → Ⅎ𝑥(℩𝑥𝜓))
14	13, 10	nfeqd 2758	. . 3 ⊢ (𝜑 → Ⅎ𝑥(℩𝑥𝜓) = 𝐵)
15	11, 14	nfbid 1820	. 2 ⊢ (𝜑 → Ⅎ𝑥(𝜒 ↔ (℩𝑥𝜓) = 𝐵))
16	1, 5, 8, 9, 10, 15	vtocldf 3229	1 ⊢ (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  ∃!weu 2458  Ⅎwnfc 2738  ℩cio 5766

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-sbc 3403  df-un 3545  df-sn 4126  df-pr 4128  df-uni 4373  df-iota 5768

This theorem is referenced by:  iota2d  5793  iota2  5794  riota2df  6531  opiota  7118