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Mirrors > Home > NFE 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 2208 | . 2 ⊢ (∃!xφ ↔ ∃z∀x(φ ↔ x = z)) | |
2 | sp 1747 | . . . . 5 ⊢ (∀x(φ ↔ x = z) → (φ ↔ x = z)) | |
3 | iotaval 4350 | . . . . . 6 ⊢ (∀x(φ ↔ x = z) → (℩xφ) = z) | |
4 | 3 | eqeq2d 2364 | . . . . 5 ⊢ (∀x(φ ↔ x = z) → (x = (℩xφ) ↔ x = z)) |
5 | 2, 4 | bitr4d 247 | . . . 4 ⊢ (∀x(φ ↔ x = z) → (φ ↔ x = (℩xφ))) |
6 | eqcom 2355 | . . . 4 ⊢ (x = (℩xφ) ↔ (℩xφ) = x) | |
7 | 5, 6 | syl6bb 252 | . . 3 ⊢ (∀x(φ ↔ x = z) → (φ ↔ (℩xφ) = x)) |
8 | 7 | exlimiv 1634 | . 2 ⊢ (∃z∀x(φ ↔ x = z) → (φ ↔ (℩xφ) = x)) |
9 | 1, 8 | sylbi 187 | 1 ⊢ (∃!xφ → (φ ↔ (℩xφ) = x)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∀wal 1540 ∃wex 1541 = wceq 1642 ∃!weu 2204 ℩cio 4337 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-rex 2620 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-un 3214 df-sn 3741 df-pr 3742 df-uni 3892 df-iota 4339 |
This theorem is referenced by: iota2df 4365 sniota 4369 |
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