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Mirrors > Home > MPE Home > Th. List > Mathboxes > iotasbc | Structured version Visualization version GIF version |
Description: Definition *14.01 in [WhiteheadRussell] p. 184. In Principia Mathematica, Russell and Whitehead define ℩ in terms of a function of (℩𝑥𝜑). Their definition differs in that a function of (℩𝑥𝜑) evaluates to "false" when there isn't a single 𝑥 that satisfies 𝜑. (Contributed by Andrew Salmon, 11-Jul-2011.) |
Ref | Expression |
---|---|
iotasbc | ⊢ (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑦]𝜓 ↔ ∃𝑦(∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbc5 3806 | . 2 ⊢ ([(℩𝑥𝜑) / 𝑦]𝜓 ↔ ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ 𝜓)) | |
2 | iotaexeu 43886 | . . . . . . 7 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V) | |
3 | eueq 3705 | . . . . . . 7 ⊢ ((℩𝑥𝜑) ∈ V ↔ ∃!𝑦 𝑦 = (℩𝑥𝜑)) | |
4 | 2, 3 | sylib 217 | . . . . . 6 ⊢ (∃!𝑥𝜑 → ∃!𝑦 𝑦 = (℩𝑥𝜑)) |
5 | eu6 2563 | . . . . . . 7 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | |
6 | iotaval 6524 | . . . . . . . . . 10 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦) | |
7 | 6 | eqcomd 2734 | . . . . . . . . 9 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → 𝑦 = (℩𝑥𝜑)) |
8 | 7 | ancri 548 | . . . . . . . 8 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (𝑦 = (℩𝑥𝜑) ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))) |
9 | 8 | eximi 1829 | . . . . . . 7 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))) |
10 | 5, 9 | sylbi 216 | . . . . . 6 ⊢ (∃!𝑥𝜑 → ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))) |
11 | eupick 2624 | . . . . . 6 ⊢ ((∃!𝑦 𝑦 = (℩𝑥𝜑) ∧ ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))) → (𝑦 = (℩𝑥𝜑) → ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))) | |
12 | 4, 10, 11 | syl2anc 582 | . . . . 5 ⊢ (∃!𝑥𝜑 → (𝑦 = (℩𝑥𝜑) → ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))) |
13 | 12, 7 | impbid1 224 | . . . 4 ⊢ (∃!𝑥𝜑 → (𝑦 = (℩𝑥𝜑) ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))) |
14 | 13 | anbi1d 629 | . . 3 ⊢ (∃!𝑥𝜑 → ((𝑦 = (℩𝑥𝜑) ∧ 𝜓) ↔ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ 𝜓))) |
15 | 14 | exbidv 1916 | . 2 ⊢ (∃!𝑥𝜑 → (∃𝑦(𝑦 = (℩𝑥𝜑) ∧ 𝜓) ↔ ∃𝑦(∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ 𝜓))) |
16 | 1, 15 | bitrid 282 | 1 ⊢ (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑦]𝜓 ↔ ∃𝑦(∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∀wal 1531 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ∃!weu 2557 Vcvv 3473 [wsbc 3778 ℩cio 6503 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2166 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3475 df-sbc 3779 df-un 3954 df-in 3956 df-ss 3966 df-sn 4633 df-pr 4635 df-uni 4913 df-iota 6505 |
This theorem is referenced by: iotasbc2 43888 iotavalb 43898 fvsb 43920 |
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