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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 2922 | . . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑉 |
| 4 | 1, 3 | nfan 1899 | . . 3 ⊢ Ⅎ𝑥(𝜑 ∧ 𝐴 ∈ 𝑉) |
| 5 | iota5f.3 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝜓 ↔ 𝑥 = 𝐴)) | |
| 6 | 4, 5 | alrimi 2213 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → ∀𝑥(𝜓 ↔ 𝑥 = 𝐴)) |
| 7 | 2 | nfeq2 2923 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦 = 𝐴 |
| 8 | eqeq2 2749 | . . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐴)) | |
| 9 | 8 | bibi2d 342 | . . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝜓 ↔ 𝑥 = 𝑦) ↔ (𝜓 ↔ 𝑥 = 𝐴))) |
| 10 | 7, 9 | albid 2222 | . . . . 5 ⊢ (𝑦 = 𝐴 → (∀𝑥(𝜓 ↔ 𝑥 = 𝑦) ↔ ∀𝑥(𝜓 ↔ 𝑥 = 𝐴))) |
| 11 | eqeq2 2749 | . . . . 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 6532 | . . . 4 ⊢ (∀𝑥(𝜓 ↔ 𝑥 = 𝑦) → (℩𝑥𝜓) = 𝑦) | |
| 14 | 12, 13 | vtoclg 3554 | . . 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 1538 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 Ⅎwnfc 2890 ℩cio 6512 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-v 3482 df-un 3956 df-ss 3968 df-sn 4627 df-pr 4629 df-uni 4908 df-iota 6514 |
| This theorem is referenced by: (None) |
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