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Theorem sbanv 1889
Description: Version of sban 1955 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 1886 . 2  |-  ( [ y  /  x ]
( ph  /\  ps )  <->  A. x ( x  =  y  ->  ( ph  /\ 
ps ) ) )
2 sb6 1886 . . . 4  |-  ( [ y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  ph )
)
3 sb6 1886 . . . 4  |-  ( [ y  /  x ] ps 
<-> 
A. x ( x  =  y  ->  ps ) )
42, 3anbi12i 460 . . 3  |-  ( ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) 
<->  ( A. x ( x  =  y  ->  ph )  /\  A. x
( x  =  y  ->  ps ) ) )
5 19.26 1481 . . 3  |-  ( A. x ( ( x  =  y  ->  ph )  /\  ( x  =  y  ->  ps ) )  <-> 
( A. x ( x  =  y  ->  ph )  /\  A. x
( x  =  y  ->  ps ) ) )
6 pm4.76 604 . . . 4  |-  ( ( ( x  =  y  ->  ph )  /\  (
x  =  y  ->  ps ) )  <->  ( x  =  y  ->  ( ph  /\ 
ps ) ) )
76albii 1470 . . 3  |-  ( A. x ( ( x  =  y  ->  ph )  /\  ( x  =  y  ->  ps ) )  <->  A. x ( x  =  y  ->  ( ph  /\ 
ps ) ) )
84, 5, 73bitr2i 208 . 2  |-  ( ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) 
<-> 
A. x ( x  =  y  ->  ( ph  /\  ps ) ) )
91, 8bitr4i 187 1  |-  ( [ y  /  x ]
( ph  /\  ps )  <->  ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105   A.wal 1351   [wsb 1762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534
This theorem depends on definitions:  df-bi 117  df-sb 1763
This theorem is referenced by:  sban  1955
  Copyright terms: Public domain W3C validator