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Theorem riota2f 7339
Description: This theorem shows a condition that allows 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 2924 . 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 483 . 2 ((𝐵𝐴𝑥 = 𝐵) → (𝜑𝜓))
92, 3, 5, 6, 8riota2df 7338 1 ((𝐵𝐴 ∧ ∃!𝑥𝐴 𝜑) → (𝜓 ↔ (𝑥𝐴 𝜑) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wnf 1786  wcel 2107  wnfc 2888  ∃!wreu 3352  crio 7313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-reu 3355  df-v 3448  df-un 3916  df-in 3918  df-ss 3928  df-sn 4588  df-pr 4590  df-uni 4867  df-iota 6449  df-riota 7314
This theorem is referenced by:  riota2  7340  riotaprop  7342  riotass2  7345  riotass  7346  riotaxfrd  7349  ttrcltr  9653  cdlemksv2  39313  cdlemkuv2  39333  cdlemk36  39379
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