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Theorem sb6a 1964
Description: Equivalence for substitution. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
sb6a  |-  ( [ y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  [ x  /  y ] ph ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem sb6a
StepHypRef Expression
1 sb6 1859 . 2  |-  ( [ y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  ph )
)
2 sbequ12 1745 . . . . 5  |-  ( y  =  x  ->  ( ph 
<->  [ x  /  y ] ph ) )
32equcoms 1685 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  [ x  /  y ] ph ) )
43pm5.74i 179 . . 3  |-  ( ( x  =  y  ->  ph )  <->  ( x  =  y  ->  [ x  /  y ] ph ) )
54albii 1447 . 2  |-  ( A. x ( x  =  y  ->  ph )  <->  A. x
( x  =  y  ->  [ x  / 
y ] ph )
)
61, 5bitri 183 1  |-  ( [ y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  [ x  /  y ] ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1330   [wsb 1736
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515
This theorem depends on definitions:  df-bi 116  df-sb 1737
This theorem is referenced by: (None)
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