Theorem iota5f 31606

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 2779	. . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑉
4	1, 3	nfan 1828	. . 3 ⊢ Ⅎ𝑥(𝜑 ∧ 𝐴 ∈ 𝑉)
5		iota5f.3	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝜓 ↔ 𝑥 = 𝐴))
6	4, 5	alrimi 2082	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → ∀𝑥(𝜓 ↔ 𝑥 = 𝐴))
7	2	nfeq2 2780	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 = 𝐴
8		eqeq2 2633	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐴))
9	8	bibi2d 332	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝜓 ↔ 𝑥 = 𝑦) ↔ (𝜓 ↔ 𝑥 = 𝐴)))
10	7, 9	albid 2090	. . . . 5 ⊢ (𝑦 = 𝐴 → (∀𝑥(𝜓 ↔ 𝑥 = 𝑦) ↔ ∀𝑥(𝜓 ↔ 𝑥 = 𝐴)))
11		eqeq2 2633	. . . . 5 ⊢ (𝑦 = 𝐴 → ((℩𝑥𝜓) = 𝑦 ↔ (℩𝑥𝜓) = 𝐴))
12	10, 11	imbi12d 334	. . . 4 ⊢ (𝑦 = 𝐴 → ((∀𝑥(𝜓 ↔ 𝑥 = 𝑦) → (℩𝑥𝜓) = 𝑦) ↔ (∀𝑥(𝜓 ↔ 𝑥 = 𝐴) → (℩𝑥𝜓) = 𝐴)))
13		iotaval 5862	. . . 4 ⊢ (∀𝑥(𝜓 ↔ 𝑥 = 𝑦) → (℩𝑥𝜓) = 𝑦)
14	12, 13	vtoclg 3266	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝜓 ↔ 𝑥 = 𝐴) → (℩𝑥𝜓) = 𝐴))
15	14	adantl 482	. 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 196   ∧ wa 384  ∀wal 1481   = wceq 1483  Ⅎwnf 1708   ∈ wcel 1990  Ⅎwnfc 2751  ℩cio 5849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-v 3202  df-sbc 3436  df-un 3579  df-sn 4178  df-pr 4180  df-uni 4437  df-iota 5851

This theorem is referenced by: (None)