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Theorem sban 1961
Description: Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
sban  |-  ( [ y  /  x ]
( ph  /\  ps )  <->  ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) )

Proof of Theorem sban
StepHypRef Expression
1 sbn 1954 . . 3  |-  ( [ y  /  x ]  -.  ( ph  ->  -.  ps )  <->  -.  [ y  /  x ] ( ph  ->  -.  ps ) )
2 sbim 1957 . . . 4  |-  ( [ y  /  x ]
( ph  ->  -.  ps ) 
<->  ( [ y  /  x ] ph  ->  [ y  /  x ]  -.  ps ) )
3 sbn 1954 . . . . 5  |-  ( [ y  /  x ]  -.  ps  <->  -.  [ y  /  x ] ps )
43imbi2i 305 . . . 4  |-  ( ( [ y  /  x ] ph  ->  [ y  /  x ]  -.  ps ) 
<->  ( [ y  /  x ] ph  ->  -.  [ y  /  x ] ps ) )
52, 4bitri 242 . . 3  |-  ( [ y  /  x ]
( ph  ->  -.  ps ) 
<->  ( [ y  /  x ] ph  ->  -.  [ y  /  x ] ps ) )
61, 5xchbinx 303 . 2  |-  ( [ y  /  x ]  -.  ( ph  ->  -.  ps )  <->  -.  ( [
y  /  x ] ph  ->  -.  [ y  /  x ] ps )
)
7 df-an 362 . . 3  |-  ( (
ph  /\  ps )  <->  -.  ( ph  ->  -.  ps ) )
87sbbii 1885 . 2  |-  ( [ y  /  x ]
( ph  /\  ps )  <->  [ y  /  x ]  -.  ( ph  ->  -.  ps ) )
9 df-an 362 . 2  |-  ( ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) 
<->  -.  ( [ y  /  x ] ph  ->  -.  [ y  /  x ] ps ) )
106, 8, 93bitr4i 270 1  |-  ( [ y  /  x ]
( ph  /\  ps )  <->  ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360   [wsb 1882
This theorem is referenced by:  sb3an  1962  sbbi  1963  sbabel  2411  cbvreu  2707  sbcan  2963  sbcang  2964  rmo3  3006  inab  3343  difab  3344  exss  4129  inopab  4723  sb5ALT  26981  2uasbanh  27020  2uasbanhVD  27377  sb5ALTVD  27379
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883
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