Theorem iota2df 5312

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 110	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵)
4	3	eqeq2d 2243	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → ((℩𝑥𝜓) = 𝑥 ↔ (℩𝑥𝜓) = 𝐵))
5	2, 4	bibi12d 235	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → ((𝜓 ↔ (℩𝑥𝜓) = 𝑥) ↔ (𝜒 ↔ (℩𝑥𝜓) = 𝐵)))
6		iota2df.2	. . 3 ⊢ (𝜑 → ∃!𝑥𝜓)
7		iota1 5301	. . 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 5288	. . . . 5 ⊢ Ⅎ𝑥(℩𝑥𝜓)
13	12	a1i 9	. . . 4 ⊢ (𝜑 → Ⅎ𝑥(℩𝑥𝜓))
14	13, 10	nfeqd 2389	. . 3 ⊢ (𝜑 → Ⅎ𝑥(℩𝑥𝜓) = 𝐵)
15	11, 14	nfbid 1636	. 2 ⊢ (𝜑 → Ⅎ𝑥(𝜒 ↔ (℩𝑥𝜓) = 𝐵))
16	1, 5, 8, 9, 10, 15	vtocldf 2855	1 ⊢ (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1397  Ⅎwnf 1508  ∃!weu 2079   ∈ wcel 2202  Ⅎwnfc 2361  ℩cio 5284

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-sn 3675  df-pr 3676  df-uni 3894  df-iota 5286

This theorem is referenced by:  iota2d  5313  iota2  5316  riota2df  5992