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Theorem riota2 5853
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 2319 . 2 𝑥𝐵
2 nfv 1528 . 2 𝑥𝜓
3 riota2.1 . 2 (𝑥 = 𝐵 → (𝜑𝜓))
41, 2, 3riota2f 5852 1 ((𝐵𝐴 ∧ ∃!𝑥𝐴 𝜑) → (𝜓 ↔ (𝑥𝐴 𝜑) = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  ∃!wreu 2457  crio 5830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-reu 2462  df-v 2740  df-sbc 2964  df-un 3134  df-sn 3599  df-pr 3600  df-uni 3811  df-iota 5179  df-riota 5831
This theorem is referenced by:  eqsupti  6995  prsrriota  7787  recriota  7889  axcaucvglemval  7896  subadd  8160  divmulap  8632  flqlelt  10276  flqbi  10290  remim  10869  resqrtcl  11038  rersqrtthlem  11039  divalgmod  11932  dfgcd3  12011  bezout  12012  oddpwdclemxy  12169  qnumdenbi  12192  ismgmid  12796  isgrpinv  12926
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