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Theorem iotabidv 5035
Description: Formula-building deduction 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 1809 . 2 (𝜑 → ∀𝑥(𝜓𝜒))
3 iotabi 5023 . 2 (∀𝑥(𝜓𝜒) → (℩𝑥𝜓) = (℩𝑥𝜒))
42, 3syl 14 1 (𝜑 → (℩𝑥𝜓) = (℩𝑥𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1294   = wceq 1296  cio 5012
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 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-rex 2376  df-uni 3676  df-iota 5014
This theorem is referenced by:  csbiotag  5042  dffv3g  5336  fveq1  5339  fveq2  5340  fvres  5364  csbfv12g  5375  fvco2  5408  riotaeqdv  5647  riotabidv  5648  riotabidva  5662  ovtposg  6062  shftval  10374  sumeq1  10898  sumeq2  10902  zsumdc  10927  isumclim3  10966  isumshft  11033
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