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Theorem sblimv 1822
Description: Version of sblim 1879 where  x and  y are distinct. (Contributed by Jim Kingdon, 19-Jan-2018.)
Hypothesis
Ref Expression
sblimv.1  |-  ( ps 
->  A. x ps )
Assertion
Ref Expression
sblimv  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  ps )
)
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem sblimv
StepHypRef Expression
1 sbimv 1821 . 2  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )
)
2 sblimv.1 . . . 4  |-  ( ps 
->  A. x ps )
32sbh 1706 . . 3  |-  ( [ y  /  x ] ps 
<->  ps )
43imbi2i 224 . 2  |-  ( ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  <->  ( [ y  /  x ] ph  ->  ps )
)
51, 4bitri 182 1  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1287   [wsb 1692
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-sb 1693
This theorem is referenced by: (None)
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