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Theorem riota1a 6996
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 529 . 2 (𝑥𝐴 → (𝜑 ↔ (𝑥𝐴𝜑)))
2 df-reu 3112 . . 3 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
3 iota1 6203 . . 3 (∃!𝑥(𝑥𝐴𝜑) → ((𝑥𝐴𝜑) ↔ (℩𝑥(𝑥𝐴𝜑)) = 𝑥))
42, 3sylbi 218 . 2 (∃!𝑥𝐴 𝜑 → ((𝑥𝐴𝜑) ↔ (℩𝑥(𝑥𝐴𝜑)) = 𝑥))
51, 4sylan9bb 510 1 ((𝑥𝐴 ∧ ∃!𝑥𝐴 𝜑) → (𝜑 ↔ (℩𝑥(𝑥𝐴𝜑)) = 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1522  wcel 2081  ∃!weu 2611  ∃!wreu 3107  cio 6187
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-rex 3111  df-reu 3112  df-v 3439  df-sbc 3707  df-un 3864  df-sn 4473  df-pr 4475  df-uni 4746  df-iota 6189
This theorem is referenced by: (None)
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