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Theorem sbcied 3811
Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.)
Hypotheses
Ref Expression
sbcied.1 (𝜑𝐴𝑉)
sbcied.2 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
Assertion
Ref Expression
sbcied (𝜑 → ([𝐴 / 𝑥]𝜓𝜒))
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝜒,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem sbcied
StepHypRef Expression
1 sbcied.1 . 2 (𝜑𝐴𝑉)
2 sbcied.2 . 2 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
3 nfv 1906 . 2 𝑥𝜑
4 nfvd 1907 . 2 (𝜑 → Ⅎ𝑥𝜒)
51, 2, 3, 4sbciedf 3810 1 (𝜑 → ([𝐴 / 𝑥]𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  [wsbc 3769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-v 3494  df-sbc 3770
This theorem is referenced by:  sbcied2  3812  sbc2iedv  3848  sbc3ie  3849  sbcralt  3852  euotd  5394  fmptsnd  6923  riota5f  7131  fpwwe2lem12  10051  fpwwe2lem13  10052  brfi1uzind  13844  opfi1uzind  13847  sbcie3s  16529  issubc  17093  gsumvalx  17874  dmdprd  19049  dprdval  19054  issrg  19186  issrng  19550  islmhm  19728  isassa  20016  isphl  20700  istmd  22610  istgp  22613  isnlm  23211  isclm  23595  iscph  23701  iscms  23875  limcfval  24397  ewlksfval  27310  sbcies  30178  abfmpeld  30327  abfmpel  30328  isomnd  30629  isorng  30799
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