MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbcthdv Structured version   Visualization version   GIF version

Theorem sbcthdv 3649
Description: Deduction version of sbcth 3648. (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 2023 . 2 (𝜑 → ∀𝑥𝜓)
3 spsbc 3646 . 2 (𝐴𝑉 → (∀𝑥𝜓[𝐴 / 𝑥]𝜓))
42, 3mpan9 503 1 ((𝜑𝐴𝑉) → [𝐴 / 𝑥]𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wal 1651  wcel 2157  [wsbc 3633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-tru 1657  df-ex 1876  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-v 3387  df-sbc 3634
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator