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Theorem sbcimdv 3465
 Description: Substitution analogue of Theorem 19.20 of [Margaris] p. 90 (alim 1729). (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 3412 . 2 ([𝐴 / 𝑥]𝜓𝐴 ∈ V)
2 sbcimdv.1 . . . . 5 (𝜑 → (𝜓𝜒))
32alrimiv 1842 . . . 4 (𝜑 → ∀𝑥(𝜓𝜒))
4 spsbc 3415 . . . 4 (𝐴 ∈ V → (∀𝑥(𝜓𝜒) → [𝐴 / 𝑥](𝜓𝜒)))
5 sbcim1 3449 . . . 4 ([𝐴 / 𝑥](𝜓𝜒) → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
63, 4, 5syl56 35 . . 3 (𝐴 ∈ V → (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)))
76com3l 87 . 2 (𝜑 → ([𝐴 / 𝑥]𝜓 → (𝐴 ∈ V → [𝐴 / 𝑥]𝜒)))
81, 7mpdi 44 1 (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1473   ∈ wcel 1977  Vcvv 3173  [wsbc 3402 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175  df-sbc 3403 This theorem is referenced by:  esum2dlem  29481
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