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Theorem sbbi 2011
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 611 . . 3  |-  ( (
ph 
<->  ps )  <->  ( ( ph  ->  ps )  /\  ( ps  ->  ph )
) )
21sbbii 1636 . 2  |-  ( [ y  /  x ]
( ph  <->  ps )  <->  [ y  /  x ] ( (
ph  ->  ps )  /\  ( ps  ->  ph )
) )
3 sbim 2005 . . . 4  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )
)
4 sbim 2005 . . . 4  |-  ( [ y  /  x ]
( ps  ->  ph )  <->  ( [ y  /  x ] ps  ->  [ y  /  x ] ph ) )
53, 4anbi12i 680 . . 3  |-  ( ( [ y  /  x ] ( ph  ->  ps )  /\  [ y  /  x ] ( ps  ->  ph ) )  <-> 
( ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  /\  ( [ y  /  x ] ps  ->  [ y  /  x ] ph ) ) )
6 sban 2009 . . 3  |-  ( [ y  /  x ]
( ( ph  ->  ps )  /\  ( ps 
->  ph ) )  <->  ( [
y  /  x ]
( ph  ->  ps )  /\  [ y  /  x ] ( ps  ->  ph ) ) )
7 dfbi2 611 . . 3  |-  ( ( [ y  /  x ] ph  <->  [ y  /  x ] ps )  <->  ( ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  /\  ( [ y  /  x ] ps  ->  [ y  /  x ] ph ) ) )
85, 6, 73bitr4i 270 . 2  |-  ( [ y  /  x ]
( ( ph  ->  ps )  /\  ( ps 
->  ph ) )  <->  ( [
y  /  x ] ph 
<->  [ y  /  x ] ps ) )
92, 8bitri 242 1  |-  ( [ y  /  x ]
( ph  <->  ps )  <->  ( [
y  /  x ] ph 
<->  [ y  /  x ] ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360   [wsb 1631
This theorem is referenced by:  sblbis  2012  sbrbis  2013  spsbbi  2017  sbco  2023  sbidm  2025  sbal  2070  sb8eu  2163  pm13.183  2910  sbcbig  3039  sb8iota  6260
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632
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