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Theorem sbcbi 28561
Description: Implication form of sbcbiiOLD 3209. sbcbi 28561 is sbcbiVD 28925 without virtual deductions and was automatically derived from sbcbiVD 28925 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  |-  ( A  e.  V  ->  ( A. x ( ph  <->  ps )  ->  ( [. A  /  x ]. ph  <->  [. A  /  x ]. ps ) ) )

Proof of Theorem sbcbi
StepHypRef Expression
1 spsbc 3165 . 2  |-  ( A  e.  V  ->  ( A. x ( ph  <->  ps )  ->  [. A  /  x ]. ( ph  <->  ps )
) )
2 sbcbig 3199 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  <->  ps )  <->  (
[. A  /  x ]. ph  <->  [. A  /  x ]. ps ) ) )
31, 2sylibd 206 1  |-  ( A  e.  V  ->  ( A. x ( ph  <->  ps )  ->  ( [. A  /  x ]. ph  <->  [. A  /  x ]. ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1549    e. wcel 1725   [.wsbc 3153
This theorem is referenced by:  onfrALTlem5  28565  trsbcVD  28926  sbcssVD  28932  onfrALTlem5VD  28934
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-sbc 3154
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