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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 6284 | . 2 ⊢ (℩𝑥𝜑) = ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} | |
2 | sbeqalb 3783 | . . . . . . . 8 ⊢ (𝑦 ∈ V → ((∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) → 𝑦 = 𝑧)) | |
3 | 2 | elv 3446 | . . . . . . 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 2033 | . . . . . . . . . 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 1917 | . . . . . . 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 2025 | . . . . 5 ⊢ (𝑦 = 𝑧 ↔ 𝑧 = 𝑦) | |
12 | 10, 11 | syl6bb 290 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ 𝑧 = 𝑦)) |
13 | 12 | alrimiv 1928 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∀𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ 𝑧 = 𝑦)) |
14 | uniabio 6297 | . . 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 | syl5eq 2845 | 1 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1536 = wceq 1538 {cab 2776 Vcvv 3441 ∪ cuni 4800 ℩cio 6281 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-sbc 3721 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-uni 4801 df-iota 6283 |
This theorem is referenced by: iotauni 6299 iota1 6301 iotaex 6304 iota4 6305 iota5 6307 iota5f 33068 iotain 41121 iotaexeu 41122 iotasbc 41123 iotaequ 41133 iotavalb 41134 pm14.24 41136 sbiota1 41138 aiotaval 43650 |
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