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Theorem riotabidv 5740
Description: Formula-building deduction for restricted iota. (Contributed by NM, 15-Sep-2011.)
Hypothesis
Ref Expression
riotabidv.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
riotabidv (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐴 𝜒))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)

Proof of Theorem riotabidv
StepHypRef Expression
1 biidd 171 . . . 4 (𝜑 → (𝑥𝐴𝑥𝐴))
2 riotabidv.1 . . . 4 (𝜑 → (𝜓𝜒))
31, 2anbi12d 465 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
43iotabidv 5117 . 2 (𝜑 → (℩𝑥(𝑥𝐴𝜓)) = (℩𝑥(𝑥𝐴𝜒)))
5 df-riota 5738 . 2 (𝑥𝐴 𝜓) = (℩𝑥(𝑥𝐴𝜓))
6 df-riota 5738 . 2 (𝑥𝐴 𝜒) = (℩𝑥(𝑥𝐴𝜒))
74, 5, 63eqtr4g 2198 1 (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐴 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1332  wcel 1481  cio 5094  crio 5737
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-uni 3745  df-iota 5096  df-riota 5738
This theorem is referenced by:  riotaeqbidv  5741  csbriotag  5750  infvalti  6917  caucvgsrlemfv  7623  axcaucvglemval  7729  axcaucvglemcau  7730  subval  7978  divvalap  8458  divfnzn  9440  flval  10076  cjval  10649  sqrtrval  10804  qnumval  11899  qdenval  11900
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