ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  riotabidv GIF version

Theorem riotabidv 5686
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 462 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
43iotabidv 5067 . 2 (𝜑 → (℩𝑥(𝑥𝐴𝜓)) = (℩𝑥(𝑥𝐴𝜒)))
5 df-riota 5684 . 2 (𝑥𝐴 𝜓) = (℩𝑥(𝑥𝐴𝜓))
6 df-riota 5684 . 2 (𝑥𝐴 𝜒) = (℩𝑥(𝑥𝐴𝜒))
74, 5, 63eqtr4g 2172 1 (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐴 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1314  wcel 1463  cio 5044  crio 5683
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-rex 2396  df-uni 3703  df-iota 5046  df-riota 5684
This theorem is referenced by:  riotaeqbidv  5687  csbriotag  5696  infvalti  6861  caucvgsrlemfv  7533  axcaucvglemval  7632  axcaucvglemcau  7633  subval  7877  divvalap  8347  divfnzn  9315  flval  9938  cjval  10510  sqrtrval  10664  qnumval  11708  qdenval  11709
  Copyright terms: Public domain W3C validator