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Theorem sbcimdv 3840
Description: Substitution analogue of Theorem 19.20 of [Margaris] p. 90 (alim 1802). (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.)
Hypothesis
Ref Expression
sbcimdv.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
sbcimdv (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)

Proof of Theorem sbcimdv
StepHypRef Expression
1 sbcex 3779 . 2 ([𝐴 / 𝑥]𝜓𝐴 ∈ V)
2 sbcimdv.1 . . . . 5 (𝜑 → (𝜓𝜒))
32alrimiv 1919 . . . 4 (𝜑 → ∀𝑥(𝜓𝜒))
4 spsbc 3782 . . . 4 (𝐴 ∈ V → (∀𝑥(𝜓𝜒) → [𝐴 / 𝑥](𝜓𝜒)))
5 sbcim1 3822 . . . 4 ([𝐴 / 𝑥](𝜓𝜒) → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
63, 4, 5syl56 36 . . 3 (𝐴 ∈ V → (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)))
76com3l 89 . 2 (𝜑 → ([𝐴 / 𝑥]𝜓 → (𝐴 ∈ V → [𝐴 / 𝑥]𝜒)))
81, 7mpdi 45 1 (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1526  wcel 2105  Vcvv 3492  [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-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:  esum2dlem  31250
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