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Theorem riota1 7115
 Description: Property of restricted iota. Compare iota1 6302. (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 3113 . . 3 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
2 iota1 6302 . . 3 (∃!𝑥(𝑥𝐴𝜑) → ((𝑥𝐴𝜑) ↔ (℩𝑥(𝑥𝐴𝜑)) = 𝑥))
31, 2sylbi 220 . 2 (∃!𝑥𝐴 𝜑 → ((𝑥𝐴𝜑) ↔ (℩𝑥(𝑥𝐴𝜑)) = 𝑥))
4 df-riota 7094 . . 3 (𝑥𝐴 𝜑) = (℩𝑥(𝑥𝐴𝜑))
54eqeq1i 2803 . 2 ((𝑥𝐴 𝜑) = 𝑥 ↔ (℩𝑥(𝑥𝐴𝜑)) = 𝑥)
63, 5bitr4di 292 1 (∃!𝑥𝐴 𝜑 → ((𝑥𝐴𝜑) ↔ (𝑥𝐴 𝜑) = 𝑥))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∃!weu 2628  ∃!wreu 3108  ℩cio 6282  ℩crio 7093 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-reu 3113  df-v 3443  df-sbc 3721  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-uni 4802  df-iota 6284  df-riota 7094 This theorem is referenced by:  nosupbnd1  33342  nosupbnd2  33344  wessf1ornlem  41854  disjinfi  41863
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