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Theorem riota1a 7410
Description: Property of iota. (Contributed by NM, 23-Aug-2011.)
Assertion
Ref Expression
riota1a ((𝑥𝐴 ∧ ∃!𝑥𝐴 𝜑) → (𝜑 ↔ (℩𝑥(𝑥𝐴𝜑)) = 𝑥))

Proof of Theorem riota1a
StepHypRef Expression
1 ibar 528 . 2 (𝑥𝐴 → (𝜑 ↔ (𝑥𝐴𝜑)))
2 df-reu 3379 . . 3 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
3 iota1 6540 . . 3 (∃!𝑥(𝑥𝐴𝜑) → ((𝑥𝐴𝜑) ↔ (℩𝑥(𝑥𝐴𝜑)) = 𝑥))
42, 3sylbi 217 . 2 (∃!𝑥𝐴 𝜑 → ((𝑥𝐴𝜑) ↔ (℩𝑥(𝑥𝐴𝜑)) = 𝑥))
51, 4sylan9bb 509 1 ((𝑥𝐴 ∧ ∃!𝑥𝐴 𝜑) → (𝜑 ↔ (℩𝑥(𝑥𝐴𝜑)) = 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  ∃!weu 2566  ∃!wreu 3376  cio 6514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-reu 3379  df-v 3480  df-un 3968  df-ss 3980  df-sn 4632  df-pr 4634  df-uni 4913  df-iota 6516
This theorem is referenced by: (None)
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