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Theorem sbcbi 42913
Description: Implication form of sbcbii 3803. sbcbi 42913 is sbcbiVD 43250 without virtual deductions and was automatically derived from sbcbiVD 43250 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 3756 . 2 (𝐴𝑉 → (∀𝑥(𝜑𝜓) → [𝐴 / 𝑥](𝜑𝜓)))
2 sbcbig 3797 . 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 1540  wcel 2107  [wsbc 3743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-sbc 3744
This theorem is referenced by:  trsbcVD  43251  sbcssgVD  43257
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