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Theorem sbcbi 27676
Description: Implication form of sbcbiiOLD 3047. sbcbi 27676 is sbcbiVD 28025 without virtual deductions and was automatically derived from sbcbiVD 28025 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 3003 . 2  |-  ( A  e.  V  ->  ( A. x ( ph  <->  ps )  ->  [. A  /  x ]. ( ph  <->  ps )
) )
2 sbcbig 3037 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  <->  ps )  <->  (
[. A  /  x ]. ph  <->  [. A  /  x ]. ps ) ) )
31, 2sylibd 205 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 176   A.wal 1527    e. wcel 1684   [.wsbc 2991
This theorem is referenced by:  onfrALTlem5  27680  trsbcVD  28026  sbcssVD  28032  onfrALTlem5VD  28034
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sbc 2992
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