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Theorem sbbi 2024
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 609 . . 3  |-  ( (
ph 
<->  ps )  <->  ( ( ph  ->  ps )  /\  ( ps  ->  ph )
) )
21sbbii 1643 . 2  |-  ( [ y  /  x ]
( ph  <->  ps )  <->  [ y  /  x ] ( (
ph  ->  ps )  /\  ( ps  ->  ph )
) )
3 sbim 2018 . . . 4  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )
)
4 sbim 2018 . . . 4  |-  ( [ y  /  x ]
( ps  ->  ph )  <->  ( [ y  /  x ] ps  ->  [ y  /  x ] ph ) )
53, 4anbi12i 678 . . 3  |-  ( ( [ y  /  x ] ( ph  ->  ps )  /\  [ y  /  x ] ( ps  ->  ph ) )  <-> 
( ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  /\  ( [ y  /  x ] ps  ->  [ y  /  x ] ph ) ) )
6 sban 2022 . . 3  |-  ( [ y  /  x ]
( ( ph  ->  ps )  /\  ( ps 
->  ph ) )  <->  ( [
y  /  x ]
( ph  ->  ps )  /\  [ y  /  x ] ( ps  ->  ph ) ) )
7 dfbi2 609 . . 3  |-  ( ( [ y  /  x ] ph  <->  [ y  /  x ] ps )  <->  ( ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  /\  ( [ y  /  x ] ps  ->  [ y  /  x ] ph ) ) )
85, 6, 73bitr4i 268 . 2  |-  ( [ y  /  x ]
( ( ph  ->  ps )  /\  ( ps 
->  ph ) )  <->  ( [
y  /  x ] ph 
<->  [ y  /  x ] ps ) )
92, 8bitri 240 1  |-  ( [ y  /  x ]
( ph  <->  ps )  <->  ( [
y  /  x ] ph 
<->  [ y  /  x ] ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   [wsb 1638
This theorem is referenced by:  sblbis  2025  sbrbis  2026  spsbbi  2030  sbco  2036  sbidm  2038  sbal  2079  sb8eu  2174  pm13.183  2921  sbcbig  3050  sb8iota  5242
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639
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