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Theorem riota2f 7412
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 2920 . 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 7411 1 ((𝐵𝐴 ∧ ∃!𝑥𝐴 𝜑) → (𝜓 ↔ (𝑥𝐴 𝜑) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wnf 1780  wcel 2106  wnfc 2888  ∃!wreu 3376  crio 7387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-reu 3379  df-v 3480  df-un 3968  df-ss 3980  df-sn 4632  df-pr 4634  df-uni 4913  df-iota 6516  df-riota 7388
This theorem is referenced by:  riota2  7413  riotaprop  7415  riotass2  7418  riotass  7419  riotaxfrd  7422  ttrcltr  9754  cdlemksv2  40830  cdlemkuv2  40850  cdlemk36  40896
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