Theorem iota5f 36585

Description: A method for computing iota. (Contributed by Scott Fenton, 13-Dec-2017.)

Hypotheses

Ref	Expression
iota5f.1	⊢ Ⅎ𝑥𝜑
iota5f.2	⊢ Ⅎ𝑥𝐴
iota5f.3	⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝜓 ↔ 𝑥 = 𝐴))

Assertion

Ref	Expression
iota5f	⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (℩𝑥𝜓) = 𝐴)

Distinct variable group:   𝑥,𝑉

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem iota5f

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iota5f.1	. . . 4 ⊢ Ⅎ𝑥𝜑
2		iota5f.2	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfel1 2938	. . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑉
4	1, 3	nfan 1931	. . 3 ⊢ Ⅎ𝑥(𝜑 ∧ 𝐴 ∈ 𝑉)
5		iota5f.3	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝜓 ↔ 𝑥 = 𝐴))
6	4, 5	alrimi 2249	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → ∀𝑥(𝜓 ↔ 𝑥 = 𝐴))
7	2	nfeq2 2939	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 = 𝐴
8		eqeq2 2772	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐴))
9	8	bibi2d 345	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝜓 ↔ 𝑥 = 𝑦) ↔ (𝜓 ↔ 𝑥 = 𝐴)))
10	7, 9	albid 2258	. . . . 5 ⊢ (𝑦 = 𝐴 → (∀𝑥(𝜓 ↔ 𝑥 = 𝑦) ↔ ∀𝑥(𝜓 ↔ 𝑥 = 𝐴)))
11		eqeq2 2772	. . . . 5 ⊢ (𝑦 = 𝐴 → ((℩𝑥𝜓) = 𝑦 ↔ (℩𝑥𝜓) = 𝐴))
12	10, 11	imbi12d 347	. . . 4 ⊢ (𝑦 = 𝐴 → ((∀𝑥(𝜓 ↔ 𝑥 = 𝑦) → (℩𝑥𝜓) = 𝑦) ↔ (∀𝑥(𝜓 ↔ 𝑥 = 𝐴) → (℩𝑥𝜓) = 𝐴)))
13		iotaval 6515	. . . 4 ⊢ (∀𝑥(𝜓 ↔ 𝑥 = 𝑦) → (℩𝑥𝜓) = 𝑦)
14	12, 13	vtoclg 3517	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝜓 ↔ 𝑥 = 𝐴) → (℩𝑥𝜓) = 𝐴))
15	14	adantl 487	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝜓 ↔ 𝑥 = 𝐴) → (℩𝑥𝜓) = 𝐴))
16	6, 15	mpd 16	1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (℩𝑥𝜓) = 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1567   = wceq 1569  Ⅎwnf 1815   ∈ wcel 2144  Ⅎwnfc 2907  ℩cio 6495

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-v 3452  df-ss 3897  df-un 4150  df-sn 4585  df-pr 4587  df-uni 4867  df-iota 6497

This theorem is used by: (None)