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Theorem sbbi 1876
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 380 . . 3  |-  ( (
ph 
<->  ps )  <->  ( ( ph  ->  ps )  /\  ( ps  ->  ph )
) )
21sbbii 1690 . 2  |-  ( [ y  /  x ]
( ph  <->  ps )  <->  [ y  /  x ] ( (
ph  ->  ps )  /\  ( ps  ->  ph )
) )
3 sbim 1870 . . . 4  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )
)
4 sbim 1870 . . . 4  |-  ( [ y  /  x ]
( ps  ->  ph )  <->  ( [ y  /  x ] ps  ->  [ y  /  x ] ph ) )
53, 4anbi12i 448 . . 3  |-  ( ( [ y  /  x ] ( ph  ->  ps )  /\  [ y  /  x ] ( ps  ->  ph ) )  <-> 
( ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  /\  ( [ y  /  x ] ps  ->  [ y  /  x ] ph ) ) )
6 sban 1872 . . 3  |-  ( [ y  /  x ]
( ( ph  ->  ps )  /\  ( ps 
->  ph ) )  <->  ( [
y  /  x ]
( ph  ->  ps )  /\  [ y  /  x ] ( ps  ->  ph ) ) )
7 dfbi2 380 . . 3  |-  ( ( [ y  /  x ] ph  <->  [ y  /  x ] ps )  <->  ( ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  /\  ( [ y  /  x ] ps  ->  [ y  /  x ] ph ) ) )
85, 6, 73bitr4i 210 . 2  |-  ( [ y  /  x ]
( ( ph  ->  ps )  /\  ( ps 
->  ph ) )  <->  ( [
y  /  x ] ph 
<->  [ y  /  x ] ps ) )
92, 8bitri 182 1  |-  ( [ y  /  x ]
( ph  <->  ps )  <->  ( [
y  /  x ] ph 
<->  [ y  /  x ] ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   [wsb 1687
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-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688
This theorem is referenced by:  sblbis  1877  sbrbis  1878  sbco  1885  sbcocom  1887  elsb3  1895  elsb4  1896  sb8eu  1956  sb8euh  1966  pm13.183  2741  sbcbig  2870  sb8iota  4925
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