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Theorem sbbi 1952
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 386 . . 3  |-  ( (
ph 
<->  ps )  <->  ( ( ph  ->  ps )  /\  ( ps  ->  ph )
) )
21sbbii 1758 . 2  |-  ( [ y  /  x ]
( ph  <->  ps )  <->  [ y  /  x ] ( (
ph  ->  ps )  /\  ( ps  ->  ph )
) )
3 sbim 1946 . . . 4  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )
)
4 sbim 1946 . . . 4  |-  ( [ y  /  x ]
( ps  ->  ph )  <->  ( [ y  /  x ] ps  ->  [ y  /  x ] ph ) )
53, 4anbi12i 457 . . 3  |-  ( ( [ y  /  x ] ( ph  ->  ps )  /\  [ y  /  x ] ( ps  ->  ph ) )  <-> 
( ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  /\  ( [ y  /  x ] ps  ->  [ y  /  x ] ph ) ) )
6 sban 1948 . . 3  |-  ( [ y  /  x ]
( ( ph  ->  ps )  /\  ( ps 
->  ph ) )  <->  ( [
y  /  x ]
( ph  ->  ps )  /\  [ y  /  x ] ( ps  ->  ph ) ) )
7 dfbi2 386 . . 3  |-  ( ( [ y  /  x ] ph  <->  [ y  /  x ] ps )  <->  ( ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  /\  ( [ y  /  x ] ps  ->  [ y  /  x ] ph ) ) )
85, 6, 73bitr4i 211 . 2  |-  ( [ y  /  x ]
( ( ph  ->  ps )  /\  ( ps 
->  ph ) )  <->  ( [
y  /  x ] ph 
<->  [ y  /  x ] ps ) )
92, 8bitri 183 1  |-  ( [ y  /  x ]
( ph  <->  ps )  <->  ( [
y  /  x ] ph 
<->  [ y  /  x ] ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   [wsb 1755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756
This theorem is referenced by:  sblbis  1953  sbrbis  1954  sbco  1961  sbcocom  1963  sb8eu  2032  sb8euh  2042  elsb1  2148  elsb2  2149  pm13.183  2868  sbcbig  3001  sb8iota  5165
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