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Mirrors > Home > MPE Home > Th. List > iota1 | Structured version Visualization version GIF version |
Description: Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.) |
Ref | Expression |
---|---|
iota1 | ⊢ (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-eu 2474 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) | |
2 | sp 2053 | . . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (𝜑 ↔ 𝑥 = 𝑧)) | |
3 | iotaval 5862 | . . . . . 6 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧) | |
4 | 3 | eqeq2d 2632 | . . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (𝑥 = (℩𝑥𝜑) ↔ 𝑥 = 𝑧)) |
5 | 2, 4 | bitr4d 271 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (𝜑 ↔ 𝑥 = (℩𝑥𝜑))) |
6 | eqcom 2629 | . . . 4 ⊢ (𝑥 = (℩𝑥𝜑) ↔ (℩𝑥𝜑) = 𝑥) | |
7 | 5, 6 | syl6bb 276 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (𝜑 ↔ (℩𝑥𝜑) = 𝑥)) |
8 | 7 | exlimiv 1858 | . 2 ⊢ (∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (𝜑 ↔ (℩𝑥𝜑) = 𝑥)) |
9 | 1, 8 | sylbi 207 | 1 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∀wal 1481 = wceq 1483 ∃wex 1704 ∃!weu 2470 ℩cio 5849 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-v 3202 df-sbc 3436 df-un 3579 df-sn 4178 df-pr 4180 df-uni 4437 df-iota 5851 |
This theorem is referenced by: iota2df 5875 sniota 5878 tz6.12-1 6210 opabiota 6261 riota1 6629 riota1a 6630 erovlem 7843 gsumval3lem2 18307 bnj1366 30900 |
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