Theorem iota2df 6345

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 487	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵)
4	3	eqeq2d 2835	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → ((℩𝑥𝜓) = 𝑥 ↔ (℩𝑥𝜓) = 𝐵))
5	2, 4	bibi12d 348	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → ((𝜓 ↔ (℩𝑥𝜓) = 𝑥) ↔ (𝜒 ↔ (℩𝑥𝜓) = 𝐵)))
6		iota2df.2	. . 3 ⊢ (𝜑 → ∃!𝑥𝜓)
7		iota1 6335	. . 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 6319	. . . . 5 ⊢ Ⅎ𝑥(℩𝑥𝜓)
13	12	a1i 11	. . . 4 ⊢ (𝜑 → Ⅎ𝑥(℩𝑥𝜓))
14	13, 10	nfeqd 2991	. . 3 ⊢ (𝜑 → Ⅎ𝑥(℩𝑥𝜓) = 𝐵)
15	11, 14	nfbid 1902	. 2 ⊢ (𝜑 → Ⅎ𝑥(𝜒 ↔ (℩𝑥𝜓) = 𝐵))
16	1, 5, 8, 9, 10, 15	vtocldf 3558	1 ⊢ (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1536  Ⅎwnf 1783   ∈ wcel 2113  ∃!weu 2652  Ⅎwnfc 2964  ℩cio 6315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-v 3499  df-sbc 3776  df-un 3944  df-in 3946  df-ss 3955  df-sn 4571  df-pr 4573  df-uni 4842  df-iota 6317

This theorem is referenced by:  iota2d  6346  iota2  6347  riota2df  7140  opiota  7760