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Theorem csbied2 3868
Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
csbied2.1 (𝜑𝐴𝑉)
csbied2.2 (𝜑𝐴 = 𝐵)
csbied2.3 ((𝜑𝑥 = 𝐵) → 𝐶 = 𝐷)
Assertion
Ref Expression
csbied2 (𝜑𝐴 / 𝑥𝐶 = 𝐷)
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝑥,𝐷
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem csbied2
StepHypRef Expression
1 csbied2.1 . 2 (𝜑𝐴𝑉)
2 id 22 . . . 4 (𝑥 = 𝐴𝑥 = 𝐴)
3 csbied2.2 . . . 4 (𝜑𝐴 = 𝐵)
42, 3sylan9eqr 2858 . . 3 ((𝜑𝑥 = 𝐴) → 𝑥 = 𝐵)
5 csbied2.3 . . 3 ((𝜑𝑥 = 𝐵) → 𝐶 = 𝐷)
64, 5syldan 594 . 2 ((𝜑𝑥 = 𝐴) → 𝐶 = 𝐷)
71, 6csbied 3867 1 (𝜑𝐴 / 𝑥𝐶 = 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  csb 3831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-v 3446  df-sbc 3724  df-csb 3832
This theorem is referenced by:  prdsval  16724  cidfval  16943  monfval  16998  idfuval  17142  isnat  17213  fucco  17228  catcval  17352  xpcval  17423  1stfval  17437  2ndfval  17440  prfval  17445  evlf2  17464  curfval  17469  hofval  17498  ipoval  17760  mntoval  30694  mgcoval  30698  poimirlem2  35058  rngcvalALTV  44582  ringcvalALTV  44628
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