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Theorem riota1a 6854
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 520 . 2 (𝑥𝐴 → (𝜑 ↔ (𝑥𝐴𝜑)))
2 df-reu 3103 . . 3 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
3 iota1 6078 . . 3 (∃!𝑥(𝑥𝐴𝜑) → ((𝑥𝐴𝜑) ↔ (℩𝑥(𝑥𝐴𝜑)) = 𝑥))
42, 3sylbi 208 . 2 (∃!𝑥𝐴 𝜑 → ((𝑥𝐴𝜑) ↔ (℩𝑥(𝑥𝐴𝜑)) = 𝑥))
51, 4sylan9bb 501 1 ((𝑥𝐴 ∧ ∃!𝑥𝐴 𝜑) → (𝜑 ↔ (℩𝑥(𝑥𝐴𝜑)) = 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wcel 2156  ∃!weu 2630  ∃!wreu 3098  cio 6062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-rex 3102  df-reu 3103  df-v 3393  df-sbc 3634  df-un 3774  df-sn 4371  df-pr 4373  df-uni 4631  df-iota 6064
This theorem is referenced by: (None)
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