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Mirrors > Home > ILE Home > Th. List > iota1 | 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 1952 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) | |
2 | sp 1447 | . . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (𝜑 ↔ 𝑥 = 𝑧)) | |
3 | iotaval 5006 | . . . . . 6 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧) | |
4 | 3 | eqeq2d 2100 | . . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (𝑥 = (℩𝑥𝜑) ↔ 𝑥 = 𝑧)) |
5 | 2, 4 | bitr4d 190 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (𝜑 ↔ 𝑥 = (℩𝑥𝜑))) |
6 | eqcom 2091 | . . . 4 ⊢ (𝑥 = (℩𝑥𝜑) ↔ (℩𝑥𝜑) = 𝑥) | |
7 | 5, 6 | syl6bb 195 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (𝜑 ↔ (℩𝑥𝜑) = 𝑥)) |
8 | 7 | exlimiv 1535 | . 2 ⊢ (∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (𝜑 ↔ (℩𝑥𝜑) = 𝑥)) |
9 | 1, 8 | sylbi 120 | 1 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1288 = wceq 1290 ∃wex 1427 ∃!weu 1949 ℩cio 4993 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-tru 1293 df-nf 1396 df-sb 1694 df-eu 1952 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-rex 2366 df-v 2624 df-sbc 2844 df-un 3006 df-sn 3458 df-pr 3459 df-uni 3662 df-iota 4995 |
This theorem is referenced by: iota2df 5019 sniota 5022 tz6.12-1 5346 riota1 5642 riota1a 5643 erovlem 6400 |
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