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Theorem sbcbiVD 28163
Description: Implication form of sbcbiiOLD 3081. 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 27797 is sbcbiVD 28163 without virtual deductions and was automatically derived from sbcbiVD 28163.
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 27836 . . . 4  |-  (. A  e.  B  ->.  A  e.  B ).
2 idn2 27885 . . . . 5  |-  (. A  e.  B ,. A. x
( ph  <->  ps )  ->.  A. x
( ph  <->  ps ) ).
3 spsbc 3037 . . . . 5  |-  ( A  e.  B  ->  ( A. x ( ph  <->  ps )  ->  [. A  /  x ]. ( ph  <->  ps )
) )
41, 2, 3e12 28008 . . . 4  |-  (. A  e.  B ,. A. x
( ph  <->  ps )  ->.  [. A  /  x ]. ( ph  <->  ps ) ).
5 sbcbig 3071 . . . . 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 28008 . . 3  |-  (. A  e.  B ,. A. x
( ph  <->  ps )  ->.  ( [. A  /  x ]. ph  <->  [. A  /  x ]. ps ) ).
87in2 27877 . 2  |-  (. A  e.  B  ->.  ( A. x
( ph  <->  ps )  ->  ( [. A  /  x ]. ph  <->  [. A  /  x ]. ps ) ) ).
98in1 27833 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 1531    e. wcel 1701   [.wsbc 3025
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-v 2824  df-sbc 3026  df-vd1 27832  df-vd2 27841
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