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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 | eu6 2569 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) | |
2 | sp 2177 | . . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (𝜑 ↔ 𝑥 = 𝑧)) | |
3 | iotaval 6515 | . . . . . 6 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧) | |
4 | 3 | eqeq2d 2744 | . . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (𝑥 = (℩𝑥𝜑) ↔ 𝑥 = 𝑧)) |
5 | 2, 4 | bitr4d 282 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (𝜑 ↔ 𝑥 = (℩𝑥𝜑))) |
6 | eqcom 2740 | . . . 4 ⊢ (𝑥 = (℩𝑥𝜑) ↔ (℩𝑥𝜑) = 𝑥) | |
7 | 5, 6 | bitrdi 287 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (𝜑 ↔ (℩𝑥𝜑) = 𝑥)) |
8 | 7 | exlimiv 1934 | . 2 ⊢ (∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (𝜑 ↔ (℩𝑥𝜑) = 𝑥)) |
9 | 1, 8 | sylbi 216 | 1 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1540 = wceq 1542 ∃wex 1782 ∃!weu 2563 ℩cio 6494 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 df-un 3954 df-in 3956 df-ss 3966 df-sn 4630 df-pr 4632 df-uni 4910 df-iota 6496 |
This theorem is referenced by: iota2df 6531 sniota 6535 tz6.12c 6914 tz6.12-1OLD 6916 opabiota 6975 riota1 7387 riota1a 7388 erovlem 8807 gsumval3lem2 19774 bnj1366 33840 funressndmafv2rn 45931 tz6.12-afv2 45948 |
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