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Theorem sbcied2 3839
Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.)
Hypotheses
Ref Expression
sbcied2.1 (𝜑𝐴𝑉)
sbcied2.2 (𝜑𝐴 = 𝐵)
sbcied2.3 ((𝜑𝑥 = 𝐵) → (𝜓𝜒))
Assertion
Ref Expression
sbcied2 (𝜑 → ([𝐴 / 𝑥]𝜓𝜒))
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝜒,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem sbcied2
StepHypRef Expression
1 sbcied2.1 . 2 (𝜑𝐴𝑉)
2 id 22 . . . 4 (𝑥 = 𝐴𝑥 = 𝐴)
3 sbcied2.2 . . . 4 (𝜑𝐴 = 𝐵)
42, 3sylan9eqr 2797 . . 3 ((𝜑𝑥 = 𝐴) → 𝑥 = 𝐵)
5 sbcied2.3 . . 3 ((𝜑𝑥 = 𝐵) → (𝜓𝜒))
64, 5syldan 591 . 2 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
71, 6sbcied 3837 1 (𝜑 → ([𝐴 / 𝑥]𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  [wsbc 3791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-sbc 3792
This theorem is referenced by:  iscat  17717  sectffval  17798  issubc  17886  isfunc  17915  cat1  18151  ismgm  18667  issgrp  18746  isnsg  19186  isrng  20172  isring  20255  isdomn  20722  islbs  21093  isassa  21894  opsrval  22082  isuhgr  29092  isushgr  29093  isupgr  29116  isumgr  29127  isuspgr  29184  isusgr  29185  isgrim  47806  isthinc  48821
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