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Theorem sbcied2 3825
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 2795 . . 3 ((𝜑𝑥 = 𝐴) → 𝑥 = 𝐵)
5 sbcied2.3 . . 3 ((𝜑𝑥 = 𝐵) → (𝜓𝜒))
64, 5syldan 592 . 2 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
71, 6sbcied 3823 1 (𝜑 → ([𝐴 / 𝑥]𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  [wsbc 3778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-sbc 3779
This theorem is referenced by:  iscat  17616  sectffval  17697  issubc  17785  isfunc  17814  cat1  18047  ismgm  18562  issgrp  18611  isnsg  19035  isring  20060  islbs  20687  isdomn  20910  isassa  21411  opsrval  21601  isuhgr  28320  isushgr  28321  isupgr  28344  isumgr  28355  isuspgr  28412  isusgr  28413  isrng  46650  isthinc  47641
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