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Theorem sbcthdv 3791
Description: Deduction version of sbcth 3790. (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 1921 . 2 (𝜑 → ∀𝑥𝜓)
3 spsbc 3788 . 2 (𝐴𝑉 → (∀𝑥𝜓[𝐴 / 𝑥]𝜓))
42, 3mpan9 507 1 ((𝜑𝐴𝑉) → [𝐴 / 𝑥]𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1528  wcel 2107  [wsbc 3775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-ext 2797
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-sbc 3776
This theorem is referenced by: (None)
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