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Theorem iotabidv 4888
 Description: Formula-building deduction rule for iota. (Contributed by NM, 20-Aug-2011.)
Hypothesis
Ref Expression
iotabidv.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
iotabidv (𝜑 → (℩𝑥𝜓) = (℩𝑥𝜒))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem iotabidv
StepHypRef Expression
1 iotabidv.1 . . 3 (𝜑 → (𝜓𝜒))
21alrimiv 1754 . 2 (𝜑 → ∀𝑥(𝜓𝜒))
3 iotabi 4876 . 2 (∀𝑥(𝜓𝜒) → (℩𝑥𝜓) = (℩𝑥𝜒))
42, 3syl 14 1 (𝜑 → (℩𝑥𝜓) = (℩𝑥𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241   = wceq 1243  ℩cio 4865 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 This theorem is referenced by:  csbiotag  4895  dffv3g  5174  fveq1  5177  fveq2  5178  fvres  5198  csbfv12g  5209  fvco2  5242  riotaeqdv  5469  riotabidv  5470  riotabidva  5484  ovtposg  5874  shftval  9426  sumeq1  9874
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