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Theorem sbcbiVD 28652
Description: Implication form of sbcbiiOLD 3047. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sbcbi 28303 is sbcbiVD 28652 without virtual deductions and was automatically derived from sbcbiVD 28652.
1::  |-  (. A  e.  B  ->.  A  e.  B ).
2::  |-  (. A  e.  B ,. A. x ( ph  <->  ps )  ->.  A. x ( ph  <->  ps ) ).
3:1,2:  |-  (. A  e.  B ,. A. x ( ph  <->  ps )  ->.  [. A  /  x ]. ( ph  <->  ps ) ).
4:1,3:  |-  (. A  e.  B ,. A. x ( ph  <->  ps )  ->.  ( [. A  /  x ]. ph  <->  [. A  /  x ]. ps ) ).
5:4:  |-  (. A  e.  B  ->.  ( A. x ( ph  <->  ps )  ->  ( [. A  /  x ]. ph  <->  [. A  /  x ]. ps ) ) ).
qed:5:  |-  ( A  e.  B  ->  ( A. x ( ph  <->  ps )  ->  ( [. A  /  x ]. ph  <->  [. A  /  x ]. ps ) ) )
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sbcbiVD  |-  ( A  e.  B  ->  ( A. x ( ph  <->  ps )  ->  ( [. A  /  x ]. ph  <->  [. A  /  x ]. ps ) ) )

Proof of Theorem sbcbiVD
StepHypRef Expression
1 idn1 28342 . . . 4  |-  (. A  e.  B  ->.  A  e.  B ).
2 idn2 28385 . . . . 5  |-  (. A  e.  B ,. A. x
( ph  <->  ps )  ->.  A. x
( ph  <->  ps ) ).
3 spsbc 3003 . . . . 5  |-  ( A  e.  B  ->  ( A. x ( ph  <->  ps )  ->  [. A  /  x ]. ( ph  <->  ps )
) )
41, 2, 3e12 28499 . . . 4  |-  (. A  e.  B ,. A. x
( ph  <->  ps )  ->.  [. A  /  x ]. ( ph  <->  ps ) ).
5 sbcbig 3037 . . . . 5  |-  ( A  e.  B  ->  ( [. A  /  x ]. ( ph  <->  ps )  <->  (
[. A  /  x ]. ph  <->  [. A  /  x ]. ps ) ) )
65biimpd 198 . . . 4  |-  ( A  e.  B  ->  ( [. A  /  x ]. ( ph  <->  ps )  ->  ( [. A  /  x ]. ph  <->  [. A  /  x ]. ps ) ) )
71, 4, 6e12 28499 . . 3  |-  (. A  e.  B ,. A. x
( ph  <->  ps )  ->.  ( [. A  /  x ]. ph  <->  [. A  /  x ]. ps ) ).
87in2 28377 . 2  |-  (. A  e.  B  ->.  ( A. x
( ph  <->  ps )  ->  ( [. A  /  x ]. ph  <->  [. A  /  x ]. ps ) ) ).
98in1 28339 1  |-  ( A  e.  B  ->  ( 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 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  df-vd1 28338  df-vd2 28347
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