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Theorem sbcthdv 2852
Description: Deduction version of sbcth 2851. (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 1802 . 2 (𝜑 → ∀𝑥𝜓)
3 spsbc 2849 . 2 (𝐴𝑉 → (∀𝑥𝜓[𝐴 / 𝑥]𝜓))
42, 3mpan9 275 1 ((𝜑𝐴𝑉) → [𝐴 / 𝑥]𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wal 1287  wcel 1438  [wsbc 2838
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621  df-sbc 2839
This theorem is referenced by: (None)
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