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Theorem riotabidv 5470
 Description: Formula-building deduction rule 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 161 . . . 4 (𝜑 → (𝑥𝐴𝑥𝐴))
2 riotabidv.1 . . . 4 (𝜑 → (𝜓𝜒))
31, 2anbi12d 442 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
43iotabidv 4888 . 2 (𝜑 → (℩𝑥(𝑥𝐴𝜓)) = (℩𝑥(𝑥𝐴𝜒)))
5 df-riota 5468 . 2 (𝑥𝐴 𝜓) = (℩𝑥(𝑥𝐴𝜓))
6 df-riota 5468 . 2 (𝑥𝐴 𝜒) = (℩𝑥(𝑥𝐴𝜒))
74, 5, 63eqtr4g 2097 1 (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐴 𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393  ℩cio 4865  ℩crio 5467 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-uni 3581  df-iota 4867  df-riota 5468 This theorem is referenced by:  riotaeqbidv  5471  csbriotag  5480  caucvgsrlemfv  6875  axcaucvglemval  6971  axcaucvglemcau  6972  subval  7203  divvalap  7653  divfnzn  8556  flval  9116  cjval  9445  sqrtrval  9598
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