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Theorem sbcthdv 3712
 Description: Deduction version of sbcth 3711. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Hypothesis
Ref Expression
sbcthdv.1 (𝜑𝜓)
Assertion
Ref Expression
sbcthdv ((𝜑𝐴𝑉) → [𝐴 / 𝑥]𝜓)
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)   𝑉(𝑥)

Proof of Theorem sbcthdv
StepHypRef Expression
1 sbcthdv.1 . . 3 (𝜑𝜓)
21alrimiv 1928 . 2 (𝜑 → ∀𝑥𝜓)
3 spsbc 3709 . 2 (𝐴𝑉 → (∀𝑥𝜓[𝐴 / 𝑥]𝜓))
42, 3mpan9 510 1 ((𝜑𝐴𝑉) → [𝐴 / 𝑥]𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1536   ∈ wcel 2111  [wsbc 3696 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-sbc 3697 This theorem is referenced by: (None)
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