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Theorem sbcbi 43290
Description: Implication form of sbcbii 3837. sbcbi 43290 is sbcbiVD 43627 without virtual deductions and was automatically derived from sbcbiVD 43627 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sbcbi (𝐴𝑉 → (∀𝑥(𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))

Proof of Theorem sbcbi
StepHypRef Expression
1 spsbc 3790 . 2 (𝐴𝑉 → (∀𝑥(𝜑𝜓) → [𝐴 / 𝑥](𝜑𝜓)))
2 sbcbig 3831 . 2 (𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
31, 2sylibd 238 1 (𝐴𝑉 → (∀𝑥(𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1539  wcel 2106  [wsbc 3777
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-sbc 3778
This theorem is referenced by:  trsbcVD  43628  sbcssgVD  43634
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