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Mathbox for Scott Fenton |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iota5f | Structured version Visualization version GIF version |
Description: A method for computing iota. (Contributed by Scott Fenton, 13-Dec-2017.) |
Ref | Expression |
---|---|
iota5f.1 | ⊢ Ⅎ𝑥𝜑 |
iota5f.2 | ⊢ Ⅎ𝑥𝐴 |
iota5f.3 | ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝜓 ↔ 𝑥 = 𝐴)) |
Ref | Expression |
---|---|
iota5f | ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (℩𝑥𝜓) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iota5f.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
2 | iota5f.2 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | nfel1 2919 | . . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑉 |
4 | 1, 3 | nfan 1896 | . . 3 ⊢ Ⅎ𝑥(𝜑 ∧ 𝐴 ∈ 𝑉) |
5 | iota5f.3 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝜓 ↔ 𝑥 = 𝐴)) | |
6 | 4, 5 | alrimi 2210 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → ∀𝑥(𝜓 ↔ 𝑥 = 𝐴)) |
7 | 2 | nfeq2 2920 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦 = 𝐴 |
8 | eqeq2 2746 | . . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐴)) | |
9 | 8 | bibi2d 342 | . . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝜓 ↔ 𝑥 = 𝑦) ↔ (𝜓 ↔ 𝑥 = 𝐴))) |
10 | 7, 9 | albid 2219 | . . . . 5 ⊢ (𝑦 = 𝐴 → (∀𝑥(𝜓 ↔ 𝑥 = 𝑦) ↔ ∀𝑥(𝜓 ↔ 𝑥 = 𝐴))) |
11 | eqeq2 2746 | . . . . 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 6533 | . . . 4 ⊢ (∀𝑥(𝜓 ↔ 𝑥 = 𝑦) → (℩𝑥𝜓) = 𝑦) | |
14 | 12, 13 | vtoclg 3553 | . . 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) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1534 = wceq 1536 Ⅎwnf 1779 ∈ wcel 2105 Ⅎwnfc 2887 ℩cio 6513 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1539 df-ex 1776 df-nf 1780 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-v 3479 df-un 3967 df-ss 3979 df-sn 4631 df-pr 4633 df-uni 4912 df-iota 6515 |
This theorem is referenced by: (None) |
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