Theorem iota2df 4911

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 108	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵)
4	3	eqeq2d 2092	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → ((℩𝑥𝜓) = 𝑥 ↔ (℩𝑥𝜓) = 𝐵))
5	2, 4	bibi12d 233	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → ((𝜓 ↔ (℩𝑥𝜓) = 𝑥) ↔ (𝜒 ↔ (℩𝑥𝜓) = 𝐵)))
6		iota2df.2	. . 3 ⊢ (𝜑 → ∃!𝑥𝜓)
7		iota1 4901	. . 3 ⊢ (∃!𝑥𝜓 → (𝜓 ↔ (℩𝑥𝜓) = 𝑥))
8	6, 7	syl 14	. 2 ⊢ (𝜑 → (𝜓 ↔ (℩𝑥𝜓) = 𝑥))
9		iota2df.4	. 2 ⊢ Ⅎ𝑥𝜑
10		iota2df.6	. 2 ⊢ (𝜑 → Ⅎ𝑥𝐵)
11		iota2df.5	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝜒)
12		nfiota1 4889	. . . . 5 ⊢ Ⅎ𝑥(℩𝑥𝜓)
13	12	a1i 9	. . . 4 ⊢ (𝜑 → Ⅎ𝑥(℩𝑥𝜓))
14	13, 10	nfeqd 2233	. . 3 ⊢ (𝜑 → Ⅎ𝑥(℩𝑥𝜓) = 𝐵)
15	11, 14	nfbid 1520	. 2 ⊢ (𝜑 → Ⅎ𝑥(𝜒 ↔ (℩𝑥𝜓) = 𝐵))
16	1, 5, 8, 9, 10, 15	vtocldf 2650	1 ⊢ (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1284  Ⅎwnf 1389   ∈ wcel 1433  ∃!weu 1941  Ⅎwnfc 2206  ℩cio 4885

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-sn 3404  df-pr 3405  df-uni 3602  df-iota 4887

This theorem is referenced by:  iota2d  4912  iota2  4913  riota2df  5508