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Theorem sblimv 1894
Description: Version of sblim 1957 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 1893 . 2  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )
)
2 sblimv.1 . . . 4  |-  ( ps 
->  A. x ps )
32sbh 1776 . . 3  |-  ( [ y  /  x ] ps 
<->  ps )
43imbi2i 226 . 2  |-  ( ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  <->  ( [ y  /  x ] ph  ->  ps )
)
51, 4bitri 184 1  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> 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-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535
This theorem depends on definitions:  df-bi 117  df-sb 1763
This theorem is referenced by: (None)
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