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Theorem csbied2 3947
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 2796 . . 3 ((𝜑𝑥 = 𝐴) → 𝑥 = 𝐵)
5 csbied2.3 . . 3 ((𝜑𝑥 = 𝐵) → 𝐶 = 𝐷)
64, 5syldan 591 . 2 ((𝜑𝑥 = 𝐴) → 𝐶 = 𝐷)
71, 6csbied 3945 1 (𝜑𝐴 / 𝑥𝐶 = 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  csb 3907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-sbc 3791  df-csb 3908
This theorem is referenced by:  prdsval  17501  cidfval  17720  monfval  17779  idfuval  17926  isnat  18001  fucco  18018  catcval  18153  xpcval  18232  1stfval  18246  2ndfval  18249  prfval  18254  evlf2  18274  curfval  18279  hofval  18308  ipoval  18587  mntoval  32956  mgcoval  32960  erlval  33244  rlocval  33245  poimirlem2  37608  rngcvalALTV  48108  ringcvalALTV  48132
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