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Theorem sbcimdv 2823
 Description: Substitution analog of Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 11-Nov-2005.)
Hypothesis
Ref Expression
sbcimdv.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
sbcimdv ((𝜑𝐴𝑉) → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)   𝑉(𝑥)

Proof of Theorem sbcimdv
StepHypRef Expression
1 sbcimdv.1 . . . . 5 (𝜑 → (𝜓𝜒))
21alrimiv 1754 . . . 4 (𝜑 → ∀𝑥(𝜓𝜒))
3 spsbc 2775 . . . 4 (𝐴𝑉 → (∀𝑥(𝜓𝜒) → [𝐴 / 𝑥](𝜓𝜒)))
42, 3syl5 28 . . 3 (𝐴𝑉 → (𝜑[𝐴 / 𝑥](𝜓𝜒)))
5 sbcimg 2804 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥](𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)))
64, 5sylibd 138 . 2 (𝐴𝑉 → (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)))
76impcom 116 1 ((𝜑𝐴𝑉) → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1241   ∈ wcel 1393  [wsbc 2764 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765 This theorem is referenced by: (None)
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