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Theorem sbbi 2104
Description: Equivalence inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
sbbi  |-  ( [ y  /  x ]
( ph  <->  ps )  <->  ( [
y  /  x ] ph 
<->  [ y  /  x ] ps ) )

Proof of Theorem sbbi
StepHypRef Expression
1 dfbi2 610 . . 3  |-  ( (
ph 
<->  ps )  <->  ( ( ph  ->  ps )  /\  ( ps  ->  ph )
) )
21sbbii 1660 . 2  |-  ( [ y  /  x ]
( ph  <->  ps )  <->  [ y  /  x ] ( (
ph  ->  ps )  /\  ( ps  ->  ph )
) )
3 sbim 2098 . . . 4  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )
)
4 sbim 2098 . . . 4  |-  ( [ y  /  x ]
( ps  ->  ph )  <->  ( [ y  /  x ] ps  ->  [ y  /  x ] ph ) )
53, 4anbi12i 679 . . 3  |-  ( ( [ y  /  x ] ( ph  ->  ps )  /\  [ y  /  x ] ( ps  ->  ph ) )  <-> 
( ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  /\  ( [ y  /  x ] ps  ->  [ y  /  x ] ph ) ) )
6 sban 2102 . . 3  |-  ( [ y  /  x ]
( ( ph  ->  ps )  /\  ( ps 
->  ph ) )  <->  ( [
y  /  x ]
( ph  ->  ps )  /\  [ y  /  x ] ( ps  ->  ph ) ) )
7 dfbi2 610 . . 3  |-  ( ( [ y  /  x ] ph  <->  [ y  /  x ] ps )  <->  ( ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  /\  ( [ y  /  x ] ps  ->  [ y  /  x ] ph ) ) )
85, 6, 73bitr4i 269 . 2  |-  ( [ y  /  x ]
( ( ph  ->  ps )  /\  ( ps 
->  ph ) )  <->  ( [
y  /  x ] ph 
<->  [ y  /  x ] ps ) )
92, 8bitri 241 1  |-  ( [ y  /  x ]
( ph  <->  ps )  <->  ( [
y  /  x ] ph 
<->  [ y  /  x ] ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   [wsb 1655
This theorem is referenced by:  sblbis  2105  sbrbis  2106  spsbbi  2110  sbco  2116  sbidm  2118  sbal  2161  sb8eu  2256  pm13.183  3019  sbcbig  3150  sb8iota  5365
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656
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