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Theorem riota2 5752
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, 10-Dec-2016.)
Hypothesis
Ref Expression
riota2.1 (𝑥 = 𝐵 → (𝜑𝜓))
Assertion
Ref Expression
riota2 ((𝐵𝐴 ∧ ∃!𝑥𝐴 𝜑) → (𝜓 ↔ (𝑥𝐴 𝜑) = 𝐵))
Distinct variable groups:   𝜓,𝑥   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem riota2
StepHypRef Expression
1 nfcv 2281 . 2 𝑥𝐵
2 nfv 1508 . 2 𝑥𝜓
3 riota2.1 . 2 (𝑥 = 𝐵 → (𝜑𝜓))
41, 2, 3riota2f 5751 1 ((𝐵𝐴 ∧ ∃!𝑥𝐴 𝜑) → (𝜓 ↔ (𝑥𝐴 𝜑) = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wcel 1480  ∃!wreu 2418  crio 5729
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-reu 2423  df-v 2688  df-sbc 2910  df-un 3075  df-sn 3533  df-pr 3534  df-uni 3737  df-iota 5088  df-riota 5730
This theorem is referenced by:  eqsupti  6883  prsrriota  7603  recriota  7705  axcaucvglemval  7712  subadd  7972  divmulap  8442  flqlelt  10056  flqbi  10070  remim  10639  resqrtcl  10808  rersqrtthlem  10809  divalgmod  11631  dfgcd3  11705  bezout  11706  oddpwdclemxy  11854  qnumdenbi  11877
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