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Theorem csbied2 3017
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 19 . . . 4 (𝑥 = 𝐴𝑥 = 𝐴)
3 csbied2.2 . . . 4 (𝜑𝐴 = 𝐵)
42, 3sylan9eqr 2172 . . 3 ((𝜑𝑥 = 𝐴) → 𝑥 = 𝐵)
5 csbied2.3 . . 3 ((𝜑𝑥 = 𝐵) → 𝐶 = 𝐷)
64, 5syldan 280 . 2 ((𝜑𝑥 = 𝐴) → 𝐶 = 𝐷)
71, 6csbied 3016 1 (𝜑𝐴 / 𝑥𝐶 = 𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1316  wcel 1465  csb 2975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-sbc 2883  df-csb 2976
This theorem is referenced by: (None)
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