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Theorem riota1 7380
Description: Property of restricted iota. Compare iota1 6511. (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 3369 . . 3 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
2 iota1 6511 . . 3 (∃!𝑥(𝑥𝐴𝜑) → ((𝑥𝐴𝜑) ↔ (℩𝑥(𝑥𝐴𝜑)) = 𝑥))
31, 2sylbi 216 . 2 (∃!𝑥𝐴 𝜑 → ((𝑥𝐴𝜑) ↔ (℩𝑥(𝑥𝐴𝜑)) = 𝑥))
4 df-riota 7358 . . 3 (𝑥𝐴 𝜑) = (℩𝑥(𝑥𝐴𝜑))
54eqeq1i 2729 . 2 ((𝑥𝐴 𝜑) = 𝑥 ↔ (℩𝑥(𝑥𝐴𝜑)) = 𝑥)
63, 5bitr4di 289 1 (∃!𝑥𝐴 𝜑 → ((𝑥𝐴𝜑) ↔ (𝑥𝐴 𝜑) = 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  ∃!weu 2554  ∃!wreu 3366  cio 6484  crio 7357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-reu 3369  df-v 3468  df-un 3946  df-in 3948  df-ss 3958  df-sn 4622  df-pr 4624  df-uni 4901  df-iota 6486  df-riota 7358
This theorem is referenced by:  nosupbnd1  27588  nosupbnd2  27590  noinfbnd1  27603  noinfbnd2  27605  wessf1ornlem  44430  disjinfi  44437
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