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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 2916 | . . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑉 |
4 | 1, 3 | nfan 1895 | . . 3 ⊢ Ⅎ𝑥(𝜑 ∧ 𝐴 ∈ 𝑉) |
5 | iota5f.3 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝜓 ↔ 𝑥 = 𝐴)) | |
6 | 4, 5 | alrimi 2202 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → ∀𝑥(𝜓 ↔ 𝑥 = 𝐴)) |
7 | 2 | nfeq2 2917 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦 = 𝐴 |
8 | eqeq2 2740 | . . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐴)) | |
9 | 8 | bibi2d 342 | . . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝜓 ↔ 𝑥 = 𝑦) ↔ (𝜓 ↔ 𝑥 = 𝐴))) |
10 | 7, 9 | albid 2211 | . . . . 5 ⊢ (𝑦 = 𝐴 → (∀𝑥(𝜓 ↔ 𝑥 = 𝑦) ↔ ∀𝑥(𝜓 ↔ 𝑥 = 𝐴))) |
11 | eqeq2 2740 | . . . . 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 6519 | . . . 4 ⊢ (∀𝑥(𝜓 ↔ 𝑥 = 𝑦) → (℩𝑥𝜓) = 𝑦) | |
14 | 12, 13 | vtoclg 3540 | . . 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 205 ∧ wa 395 ∀wal 1532 = wceq 1534 Ⅎwnf 1778 ∈ wcel 2099 Ⅎwnfc 2879 ℩cio 6498 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-v 3473 df-un 3952 df-in 3954 df-ss 3964 df-sn 4630 df-pr 4632 df-uni 4909 df-iota 6500 |
This theorem is referenced by: (None) |
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