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Mirrors > Home > MPE Home > Th. List > iotaval | Structured version Visualization version GIF version |
Description: Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.) |
Ref | Expression |
---|---|
iotaval | ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfiota2 6356 | . 2 ⊢ (℩𝑥𝜑) = ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} | |
2 | sbeqalb 3777 | . . . . . . . 8 ⊢ (𝑦 ∈ V → ((∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) → 𝑦 = 𝑧)) | |
3 | 2 | elv 3426 | . . . . . . 7 ⊢ ((∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) → 𝑦 = 𝑧) |
4 | 3 | ex 416 | . . . . . 6 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → 𝑦 = 𝑧)) |
5 | equequ2 2034 | . . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑧)) | |
6 | 5 | bibi2d 346 | . . . . . . . . 9 ⊢ (𝑦 = 𝑧 → ((𝜑 ↔ 𝑥 = 𝑦) ↔ (𝜑 ↔ 𝑥 = 𝑧))) |
7 | 6 | biimpd 232 | . . . . . . . 8 ⊢ (𝑦 = 𝑧 → ((𝜑 ↔ 𝑥 = 𝑦) → (𝜑 ↔ 𝑥 = 𝑧))) |
8 | 7 | alimdv 1924 | . . . . . . 7 ⊢ (𝑦 = 𝑧 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∀𝑥(𝜑 ↔ 𝑥 = 𝑧))) |
9 | 8 | com12 32 | . . . . . 6 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (𝑦 = 𝑧 → ∀𝑥(𝜑 ↔ 𝑥 = 𝑧))) |
10 | 4, 9 | impbid 215 | . . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ 𝑦 = 𝑧)) |
11 | equcom 2026 | . . . . 5 ⊢ (𝑦 = 𝑧 ↔ 𝑧 = 𝑦) | |
12 | 10, 11 | bitrdi 290 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ 𝑧 = 𝑦)) |
13 | 12 | alrimiv 1935 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∀𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ 𝑧 = 𝑦)) |
14 | uniabio 6370 | . . 3 ⊢ (∀𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ 𝑧 = 𝑦) → ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} = 𝑦) | |
15 | 13, 14 | syl 17 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} = 𝑦) |
16 | 1, 15 | eqtrid 2790 | 1 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1541 = wceq 1543 {cab 2715 Vcvv 3420 ∪ cuni 4833 ℩cio 6353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-nf 1792 df-sb 2072 df-clab 2716 df-cleq 2730 df-clel 2817 df-v 3422 df-un 3885 df-in 3887 df-ss 3897 df-sn 4556 df-pr 4558 df-uni 4834 df-iota 6355 |
This theorem is referenced by: iotauni 6372 iota1 6374 iotaex 6377 iota4 6378 iota5 6380 iota5f 33407 iotain 41736 iotaexeu 41737 iotasbc 41738 iotaequ 41748 iotavalb 41749 pm14.24 41751 sbiota1 41753 aiotaval 44287 |
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