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Theorem sbanv 1811
Description: Version of sban 1871 where  x and  y are distinct. (Contributed by Jim Kingdon, 24-Dec-2017.)
Assertion
Ref Expression
sbanv  |-  ( [ y  /  x ]
( ph  /\  ps )  <->  ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem sbanv
StepHypRef Expression
1 sb6 1808 . 2  |-  ( [ y  /  x ]
( ph  /\  ps )  <->  A. x ( x  =  y  ->  ( ph  /\ 
ps ) ) )
2 sb6 1808 . . . 4  |-  ( [ y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  ph )
)
3 sb6 1808 . . . 4  |-  ( [ y  /  x ] ps 
<-> 
A. x ( x  =  y  ->  ps ) )
42, 3anbi12i 448 . . 3  |-  ( ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) 
<->  ( A. x ( x  =  y  ->  ph )  /\  A. x
( x  =  y  ->  ps ) ) )
5 19.26 1411 . . 3  |-  ( A. x ( ( x  =  y  ->  ph )  /\  ( x  =  y  ->  ps ) )  <-> 
( A. x ( x  =  y  ->  ph )  /\  A. x
( x  =  y  ->  ps ) ) )
6 pm4.76 569 . . . 4  |-  ( ( ( x  =  y  ->  ph )  /\  (
x  =  y  ->  ps ) )  <->  ( x  =  y  ->  ( ph  /\ 
ps ) ) )
76albii 1400 . . 3  |-  ( A. x ( ( x  =  y  ->  ph )  /\  ( x  =  y  ->  ps ) )  <->  A. x ( x  =  y  ->  ( ph  /\ 
ps ) ) )
84, 5, 73bitr2i 206 . 2  |-  ( ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) 
<-> 
A. x ( x  =  y  ->  ( ph  /\  ps ) ) )
91, 8bitr4i 185 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   A.wal 1283   [wsb 1686
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-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-sb 1687
This theorem is referenced by:  sban  1871
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