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Theorem riota1 7124
Description: Property of restricted iota. Compare iota1 6325. (Contributed by Mario Carneiro, 15-Oct-2016.)
Assertion
Ref Expression
riota1 (∃!𝑥𝐴 𝜑 → ((𝑥𝐴𝜑) ↔ (𝑥𝐴 𝜑) = 𝑥))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem riota1
StepHypRef Expression
1 df-reu 3142 . . 3 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
2 iota1 6325 . . 3 (∃!𝑥(𝑥𝐴𝜑) → ((𝑥𝐴𝜑) ↔ (℩𝑥(𝑥𝐴𝜑)) = 𝑥))
31, 2sylbi 218 . 2 (∃!𝑥𝐴 𝜑 → ((𝑥𝐴𝜑) ↔ (℩𝑥(𝑥𝐴𝜑)) = 𝑥))
4 df-riota 7103 . . 3 (𝑥𝐴 𝜑) = (℩𝑥(𝑥𝐴𝜑))
54eqeq1i 2823 . 2 ((𝑥𝐴 𝜑) = 𝑥 ↔ (℩𝑥(𝑥𝐴𝜑)) = 𝑥)
63, 5syl6bbr 290 1 (∃!𝑥𝐴 𝜑 → ((𝑥𝐴𝜑) ↔ (𝑥𝐴 𝜑) = 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  ∃!weu 2646  ∃!wreu 3137  cio 6305  crio 7102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-reu 3142  df-v 3494  df-sbc 3770  df-un 3938  df-sn 4558  df-pr 4560  df-uni 4831  df-iota 6307  df-riota 7103
This theorem is referenced by:  nosupbnd1  33111  nosupbnd2  33113  wessf1ornlem  41321  disjinfi  41330
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