Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > iota2df | Structured version Visualization version GIF version |
Description: A condition that allows us 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 485 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵) | |
4 | 3 | eqeq2d 2829 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → ((℩𝑥𝜓) = 𝑥 ↔ (℩𝑥𝜓) = 𝐵)) |
5 | 2, 4 | bibi12d 347 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → ((𝜓 ↔ (℩𝑥𝜓) = 𝑥) ↔ (𝜒 ↔ (℩𝑥𝜓) = 𝐵))) |
6 | iota2df.2 | . . 3 ⊢ (𝜑 → ∃!𝑥𝜓) | |
7 | iota1 6325 | . . 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 6309 | . . . . 5 ⊢ Ⅎ𝑥(℩𝑥𝜓) | |
13 | 12 | a1i 11 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥(℩𝑥𝜓)) |
14 | 13, 10 | nfeqd 2985 | . . 3 ⊢ (𝜑 → Ⅎ𝑥(℩𝑥𝜓) = 𝐵) |
15 | 11, 14 | nfbid 1894 | . 2 ⊢ (𝜑 → Ⅎ𝑥(𝜒 ↔ (℩𝑥𝜓) = 𝐵)) |
16 | 1, 5, 8, 9, 10, 15 | vtocldf 3553 | 1 ⊢ (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 Ⅎwnf 1775 ∈ wcel 2105 ∃!weu 2646 Ⅎwnfc 2958 ℩cio 6305 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-v 3494 df-sbc 3770 df-un 3938 df-sn 4558 df-pr 4560 df-uni 4831 df-iota 6307 |
This theorem is referenced by: iota2d 6336 iota2 6337 riota2df 7126 opiota 7746 |
Copyright terms: Public domain | W3C validator |