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Theorem riota2f 5827
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 2323 . 2 𝑥 𝐵𝐴
31a1i 9 . 2 (𝐵𝐴𝑥𝐵)
4 riota2f.2 . . 3 𝑥𝜓
54a1i 9 . 2 (𝐵𝐴 → Ⅎ𝑥𝜓)
6 id 19 . 2 (𝐵𝐴𝐵𝐴)
7 riota2f.3 . . 3 (𝑥 = 𝐵 → (𝜑𝜓))
87adantl 275 . 2 ((𝐵𝐴𝑥 = 𝐵) → (𝜑𝜓))
92, 3, 5, 6, 8riota2df 5826 1 ((𝐵𝐴 ∧ ∃!𝑥𝐴 𝜑) → (𝜓 ↔ (𝑥𝐴 𝜑) = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1348  wnf 1453  wcel 2141  wnfc 2299  ∃!wreu 2450  crio 5805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-reu 2455  df-v 2732  df-sbc 2956  df-un 3125  df-sn 3587  df-pr 3588  df-uni 3795  df-iota 5158  df-riota 5806
This theorem is referenced by:  riota2  5828  riotaprop  5829  riotass2  5832  riotass  5833
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