Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  riota2f Structured version   Visualization version   GIF version

Theorem riota2f 7137
 Description: This theorem shows a condition that allows us to represent a descriptor with a class expression 𝐵. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
riota2f.1 𝑥𝐵
riota2f.2 𝑥𝜓
riota2f.3 (𝑥 = 𝐵 → (𝜑𝜓))
Assertion
Ref Expression
riota2f ((𝐵𝐴 ∧ ∃!𝑥𝐴 𝜑) → (𝜓 ↔ (𝑥𝐴 𝜑) = 𝐵))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝐵(𝑥)

Proof of Theorem riota2f
StepHypRef Expression
1 riota2f.1 . . 3 𝑥𝐵
21nfel1 2935 . 2 𝑥 𝐵𝐴
31a1i 11 . 2 (𝐵𝐴𝑥𝐵)
4 riota2f.2 . . 3 𝑥𝜓
54a1i 11 . 2 (𝐵𝐴 → Ⅎ𝑥𝜓)
6 id 22 . 2 (𝐵𝐴𝐵𝐴)
7 riota2f.3 . . 3 (𝑥 = 𝐵 → (𝜑𝜓))
87adantl 485 . 2 ((𝐵𝐴𝑥 = 𝐵) → (𝜑𝜓))
92, 3, 5, 6, 8riota2df 7136 1 ((𝐵𝐴 ∧ ∃!𝑥𝐴 𝜑) → (𝜓 ↔ (𝑥𝐴 𝜑) = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  Ⅎwnf 1785   ∈ wcel 2111  Ⅎwnfc 2899  ∃!wreu 3072  ℩crio 7112 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-reu 3077  df-v 3411  df-sbc 3699  df-un 3865  df-in 3867  df-ss 3877  df-sn 4526  df-pr 4528  df-uni 4802  df-iota 6298  df-riota 7113 This theorem is referenced by:  riota2  7138  riotaprop  7140  riotass2  7143  riotass  7144  riotaxfrd  7147  cdlemksv2  38449  cdlemkuv2  38469  cdlemk36  38515
 Copyright terms: Public domain W3C validator