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Theorem riota1 7332
Description: Property of restricted iota. Compare iota1 6471. (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 3353 . . 3 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
2 iota1 6471 . . 3 (∃!𝑥(𝑥𝐴𝜑) → ((𝑥𝐴𝜑) ↔ (℩𝑥(𝑥𝐴𝜑)) = 𝑥))
31, 2sylbi 216 . 2 (∃!𝑥𝐴 𝜑 → ((𝑥𝐴𝜑) ↔ (℩𝑥(𝑥𝐴𝜑)) = 𝑥))
4 df-riota 7310 . . 3 (𝑥𝐴 𝜑) = (℩𝑥(𝑥𝐴𝜑))
54eqeq1i 2741 . 2 ((𝑥𝐴 𝜑) = 𝑥 ↔ (℩𝑥(𝑥𝐴𝜑)) = 𝑥)
63, 5bitr4di 288 1 (∃!𝑥𝐴 𝜑 → ((𝑥𝐴𝜑) ↔ (𝑥𝐴 𝜑) = 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  ∃!weu 2566  ∃!wreu 3350  cio 6444  crio 7309
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-reu 3353  df-v 3446  df-un 3914  df-in 3916  df-ss 3926  df-sn 4586  df-pr 4588  df-uni 4865  df-iota 6446  df-riota 7310
This theorem is referenced by:  nosupbnd1  27058  nosupbnd2  27060  noinfbnd1  27073  noinfbnd2  27075  wessf1ornlem  43377  disjinfi  43386
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