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Theorem riotabidv 5497
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 165 . . . 4 (𝜑 → (𝑥𝐴𝑥𝐴))
2 riotabidv.1 . . . 4 (𝜑 → (𝜓𝜒))
31, 2anbi12d 450 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
43iotabidv 4915 . 2 (𝜑 → (℩𝑥(𝑥𝐴𝜓)) = (℩𝑥(𝑥𝐴𝜒)))
5 df-riota 5495 . 2 (𝑥𝐴 𝜓) = (℩𝑥(𝑥𝐴𝜓))
6 df-riota 5495 . 2 (𝑥𝐴 𝜒) = (℩𝑥(𝑥𝐴𝜒))
74, 5, 63eqtr4g 2113 1 (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐴 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  cio 4892  crio 5494
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-uni 3608  df-iota 4894  df-riota 5495
This theorem is referenced by:  riotaeqbidv  5498  csbriotag  5507  caucvgsrlemfv  6932  axcaucvglemval  7028  axcaucvglemcau  7029  subval  7265  divvalap  7726  divfnzn  8652  flval  9223  cjval  9672  sqrtrval  9826
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