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| Mirrors > Home > MPE Home > Th. List > iota2df | Structured version Visualization version GIF version | ||
| Description: A condition that allows to represent "the unique element such that 𝜑 " with a class expression 𝐴. (Contributed by NM, 30-Dec-2014.) | 
| Ref | Expression | 
|---|---|
| iota2df.1 | ⊢ (𝜑 → 𝐵 ∈ 𝑉) | 
| iota2df.2 | ⊢ (𝜑 → ∃!𝑥𝜓) | 
| iota2df.3 | ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒)) | 
| iota2df.4 | ⊢ Ⅎ𝑥𝜑 | 
| iota2df.5 | ⊢ (𝜑 → Ⅎ𝑥𝜒) | 
| iota2df.6 | ⊢ (𝜑 → Ⅎ𝑥𝐵) | 
| Ref | Expression | 
|---|---|
| iota2df | ⊢ (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | iota2df.1 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
| 2 | iota2df.3 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒)) | |
| 3 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵) | |
| 4 | 3 | eqeq2d 2748 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → ((℩𝑥𝜓) = 𝑥 ↔ (℩𝑥𝜓) = 𝐵)) | 
| 5 | 2, 4 | bibi12d 345 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → ((𝜓 ↔ (℩𝑥𝜓) = 𝑥) ↔ (𝜒 ↔ (℩𝑥𝜓) = 𝐵))) | 
| 6 | iota2df.2 | . . 3 ⊢ (𝜑 → ∃!𝑥𝜓) | |
| 7 | iota1 6538 | . . 3 ⊢ (∃!𝑥𝜓 → (𝜓 ↔ (℩𝑥𝜓) = 𝑥)) | |
| 8 | 6, 7 | syl 17 | . 2 ⊢ (𝜑 → (𝜓 ↔ (℩𝑥𝜓) = 𝑥)) | 
| 9 | iota2df.4 | . 2 ⊢ Ⅎ𝑥𝜑 | |
| 10 | iota2df.6 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
| 11 | iota2df.5 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
| 12 | nfiota1 6516 | . . . . 5 ⊢ Ⅎ𝑥(℩𝑥𝜓) | |
| 13 | 12 | a1i 11 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥(℩𝑥𝜓)) | 
| 14 | 13, 10 | nfeqd 2916 | . . 3 ⊢ (𝜑 → Ⅎ𝑥(℩𝑥𝜓) = 𝐵) | 
| 15 | 11, 14 | nfbid 1902 | . 2 ⊢ (𝜑 → Ⅎ𝑥(𝜒 ↔ (℩𝑥𝜓) = 𝐵)) | 
| 16 | 1, 5, 8, 9, 10, 15 | vtocldf 3560 | 1 ⊢ (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 ∃!weu 2568 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-3an 1089 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 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: iota2d 6549 iota2 6550 riota2df 7411 opiota 8084 | 
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