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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 2746 | . . 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 6540 | . . 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 6518 | . . . . 5 ⊢ Ⅎ𝑥(℩𝑥𝜓) | |
13 | 12 | a1i 11 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥(℩𝑥𝜓)) |
14 | 13, 10 | nfeqd 2914 | . . 3 ⊢ (𝜑 → Ⅎ𝑥(℩𝑥𝜓) = 𝐵) |
15 | 11, 14 | nfbid 1900 | . 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 1537 Ⅎwnf 1780 ∈ wcel 2106 ∃!weu 2566 Ⅎwnfc 2888 ℩cio 6514 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-v 3480 df-un 3968 df-ss 3980 df-sn 4632 df-pr 4634 df-uni 4913 df-iota 6516 |
This theorem is referenced by: iota2d 6551 iota2 6552 riota2df 7411 opiota 8083 |
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