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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 3676 | . 2 ⊢ ([(℩𝑥𝜑) / 𝑦]𝜓 ↔ ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ 𝜓)) | |
2 | iotaexeu 39556 | . . . . . . 7 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V) | |
3 | eueq 3588 | . . . . . . 7 ⊢ ((℩𝑥𝜑) ∈ V ↔ ∃!𝑦 𝑦 = (℩𝑥𝜑)) | |
4 | 2, 3 | sylib 210 | . . . . . 6 ⊢ (∃!𝑥𝜑 → ∃!𝑦 𝑦 = (℩𝑥𝜑)) |
5 | eu6 2591 | . . . . . . 7 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | |
6 | iotaval 6110 | . . . . . . . . . 10 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦) | |
7 | 6 | eqcomd 2783 | . . . . . . . . 9 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → 𝑦 = (℩𝑥𝜑)) |
8 | 7 | ancri 545 | . . . . . . . 8 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (𝑦 = (℩𝑥𝜑) ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))) |
9 | 8 | eximi 1878 | . . . . . . 7 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))) |
10 | 5, 9 | sylbi 209 | . . . . . 6 ⊢ (∃!𝑥𝜑 → ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))) |
11 | eupick 2662 | . . . . . 6 ⊢ ((∃!𝑦 𝑦 = (℩𝑥𝜑) ∧ ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))) → (𝑦 = (℩𝑥𝜑) → ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))) | |
12 | 4, 10, 11 | syl2anc 579 | . . . . 5 ⊢ (∃!𝑥𝜑 → (𝑦 = (℩𝑥𝜑) → ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))) |
13 | 12, 7 | impbid1 217 | . . . 4 ⊢ (∃!𝑥𝜑 → (𝑦 = (℩𝑥𝜑) ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))) |
14 | 13 | anbi1d 623 | . . 3 ⊢ (∃!𝑥𝜑 → ((𝑦 = (℩𝑥𝜑) ∧ 𝜓) ↔ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ 𝜓))) |
15 | 14 | exbidv 1964 | . 2 ⊢ (∃!𝑥𝜑 → (∃𝑦(𝑦 = (℩𝑥𝜑) ∧ 𝜓) ↔ ∃𝑦(∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ 𝜓))) |
16 | 1, 15 | syl5bb 275 | 1 ⊢ (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑦]𝜓 ↔ ∃𝑦(∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∀wal 1599 = wceq 1601 ∃wex 1823 ∈ wcel 2106 ∃!weu 2585 Vcvv 3397 [wsbc 3651 ℩cio 6097 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-rex 3095 df-v 3399 df-sbc 3652 df-un 3796 df-sn 4398 df-pr 4400 df-uni 4672 df-iota 6099 |
This theorem is referenced by: iotasbc2 39558 iotavalb 39568 fvsb 39592 |
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