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Theorem sbcbiVD 28990
Description: Implication form of sbcbiiOLD 3219. 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 28626 is sbcbiVD 28990 without virtual deductions and was automatically derived from sbcbiVD 28990.
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 28667 . . . 4  |-  (. A  e.  B  ->.  A  e.  B ).
2 idn2 28716 . . . . 5  |-  (. A  e.  B ,. A. x
( ph  <->  ps )  ->.  A. x
( ph  <->  ps ) ).
3 spsbc 3175 . . . . 5  |-  ( A  e.  B  ->  ( A. x ( ph  <->  ps )  ->  [. A  /  x ]. ( ph  <->  ps )
) )
41, 2, 3e12 28838 . . . 4  |-  (. A  e.  B ,. A. x
( ph  <->  ps )  ->.  [. A  /  x ]. ( ph  <->  ps ) ).
5 sbcbig 3209 . . . . 5  |-  ( A  e.  B  ->  ( [. A  /  x ]. ( ph  <->  ps )  <->  (
[. A  /  x ]. ph  <->  [. A  /  x ]. ps ) ) )
65biimpd 200 . . . 4  |-  ( A  e.  B  ->  ( [. A  /  x ]. ( ph  <->  ps )  ->  ( [. A  /  x ]. ph  <->  [. A  /  x ]. ps ) ) )
71, 4, 6e12 28838 . . 3  |-  (. A  e.  B ,. A. x
( ph  <->  ps )  ->.  ( [. A  /  x ]. ph  <->  [. A  /  x ]. ps ) ).
87in2 28708 . 2  |-  (. A  e.  B  ->.  ( A. x
( ph  <->  ps )  ->  ( [. A  /  x ]. ph  <->  [. A  /  x ]. ps ) ) ).
98in1 28664 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 178   A.wal 1550    e. wcel 1726   [.wsbc 3163
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-sbc 3164  df-vd1 28663  df-vd2 28672
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