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Theorem riota2f 7393
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 2918 . 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 481 . 2 ((𝐵𝐴𝑥 = 𝐵) → (𝜑𝜓))
92, 3, 5, 6, 8riota2df 7392 1 ((𝐵𝐴 ∧ ∃!𝑥𝐴 𝜑) → (𝜓 ↔ (𝑥𝐴 𝜑) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1540  wnf 1784  wcel 2105  wnfc 2882  ∃!wreu 3373  crio 7367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-reu 3376  df-v 3475  df-un 3953  df-in 3955  df-ss 3965  df-sn 4629  df-pr 4631  df-uni 4909  df-iota 6495  df-riota 7368
This theorem is referenced by:  riota2  7394  riotaprop  7396  riotass2  7399  riotass  7400  riotaxfrd  7403  ttrcltr  9717  cdlemksv2  40182  cdlemkuv2  40202  cdlemk36  40248
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