Theorem iota5f 35227

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 2913	. . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑉
4	1, 3	nfan 1894	. . 3 ⊢ Ⅎ𝑥(𝜑 ∧ 𝐴 ∈ 𝑉)
5		iota5f.3	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝜓 ↔ 𝑥 = 𝐴))
6	4, 5	alrimi 2198	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → ∀𝑥(𝜓 ↔ 𝑥 = 𝐴))
7	2	nfeq2 2914	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 = 𝐴
8		eqeq2 2738	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐴))
9	8	bibi2d 342	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝜓 ↔ 𝑥 = 𝑦) ↔ (𝜓 ↔ 𝑥 = 𝐴)))
10	7, 9	albid 2207	. . . . 5 ⊢ (𝑦 = 𝐴 → (∀𝑥(𝜓 ↔ 𝑥 = 𝑦) ↔ ∀𝑥(𝜓 ↔ 𝑥 = 𝐴)))
11		eqeq2 2738	. . . . 5 ⊢ (𝑦 = 𝐴 → ((℩𝑥𝜓) = 𝑦 ↔ (℩𝑥𝜓) = 𝐴))
12	10, 11	imbi12d 344	. . . 4 ⊢ (𝑦 = 𝐴 → ((∀𝑥(𝜓 ↔ 𝑥 = 𝑦) → (℩𝑥𝜓) = 𝑦) ↔ (∀𝑥(𝜓 ↔ 𝑥 = 𝐴) → (℩𝑥𝜓) = 𝐴)))
13		iotaval 6507	. . . 4 ⊢ (∀𝑥(𝜓 ↔ 𝑥 = 𝑦) → (℩𝑥𝜓) = 𝑦)
14	12, 13	vtoclg 3537	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝜓 ↔ 𝑥 = 𝐴) → (℩𝑥𝜓) = 𝐴))
15	14	adantl 481	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝜓 ↔ 𝑥 = 𝐴) → (℩𝑥𝜓) = 𝐴))
16	6, 15	mpd 15	1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (℩𝑥𝜓) = 𝐴)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1531   = wceq 1533  Ⅎwnf 1777   ∈ wcel 2098  Ⅎwnfc 2877  ℩cio 6486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-v 3470  df-un 3948  df-in 3950  df-ss 3960  df-sn 4624  df-pr 4626  df-uni 4903  df-iota 6488

This theorem is referenced by: (None)