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Theorem iota1 5834
Description: Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
Assertion
Ref Expression
iota1 (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))

Proof of Theorem iota1
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-eu 2473 . 2 (∃!𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
2 sp 2051 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑧) → (𝜑𝑥 = 𝑧))
3 iotaval 5831 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧)
43eqeq2d 2631 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑧) → (𝑥 = (℩𝑥𝜑) ↔ 𝑥 = 𝑧))
52, 4bitr4d 271 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) → (𝜑𝑥 = (℩𝑥𝜑)))
6 eqcom 2628 . . . 4 (𝑥 = (℩𝑥𝜑) ↔ (℩𝑥𝜑) = 𝑥)
75, 6syl6bb 276 . . 3 (∀𝑥(𝜑𝑥 = 𝑧) → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
87exlimiv 1855 . 2 (∃𝑧𝑥(𝜑𝑥 = 𝑧) → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
91, 8sylbi 207 1 (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1478   = wceq 1480  wex 1701  ∃!weu 2469  cio 5818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2914  df-v 3192  df-sbc 3423  df-un 3565  df-sn 4156  df-pr 4158  df-uni 4410  df-iota 5820
This theorem is referenced by:  iota2df  5844  sniota  5847  tz6.12-1  6177  opabiota  6228  riota1  6594  riota1a  6595  erovlem  7803  gsumval3lem2  18247  bnj1366  30661
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