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Theorem spsbbi 1844
Description: Specialization of biconditional. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.)
Assertion
Ref Expression
spsbbi  |-  ( A. x ( ph  <->  ps )  ->  ( [ y  /  x ] ph  <->  [ y  /  x ] ps )
)

Proof of Theorem spsbbi
StepHypRef Expression
1 spsbim 1843 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( [
y  /  x ] ph  ->  [ y  /  x ] ps ) )
2 spsbim 1843 . . 3  |-  ( A. x ( ps  ->  ph )  ->  ( [
y  /  x ] ps  ->  [ y  /  x ] ph ) )
31, 2anim12i 338 . 2  |-  ( ( A. x ( ph  ->  ps )  /\  A. x ( ps  ->  ph ) )  ->  (
( [ y  /  x ] ph  ->  [ y  /  x ] ps )  /\  ( [ y  /  x ] ps  ->  [ y  /  x ] ph ) ) )
4 albiim 1487 . 2  |-  ( A. x ( ph  <->  ps )  <->  ( A. x ( ph  ->  ps )  /\  A. x ( ps  ->  ph ) ) )
5 dfbi2 388 . 2  |-  ( ( [ y  /  x ] ph  <->  [ y  /  x ] ps )  <->  ( ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  /\  ( [ y  /  x ] ps  ->  [ y  /  x ] ph ) ) )
63, 4, 53imtr4i 201 1  |-  ( A. x ( ph  <->  ps )  ->  ( [ y  /  x ] ph  <->  [ y  /  x ] ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105   A.wal 1351   [wsb 1762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-ial 1534
This theorem depends on definitions:  df-bi 117  df-sb 1763
This theorem is referenced by:  sbbidh  1845  sbbid  1846  relelfvdm  5546
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