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Theorem sb4bor 1732
Description: Simplified definition of substitution when variables are distinct, expressed via disjunction. (Contributed by Jim Kingdon, 18-Mar-2018.)
Assertion
Ref Expression
sb4bor  |-  ( A. x  x  =  y  \/  A. x ( [ y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  ph )
) )

Proof of Theorem sb4bor
StepHypRef Expression
1 sb4or 1730 . 2  |-  ( A. x  x  =  y  \/  A. x ( [ y  /  x ] ph  ->  A. x ( x  =  y  ->  ph )
) )
2 sb2 1666 . . . . 5  |-  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ph )
3 df-bi 114 . . . . . 6  |-  ( ( ( [ y  /  x ] ph  <->  A. x
( x  =  y  ->  ph ) )  -> 
( ( [ y  /  x ] ph  ->  A. x ( x  =  y  ->  ph )
)  /\  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ph ) ) )  /\  ( ( ( [ y  /  x ] ph  ->  A. x
( x  =  y  ->  ph ) )  /\  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ph ) )  -> 
( [ y  /  x ] ph  <->  A. x
( x  =  y  ->  ph ) ) ) )
43simpri 110 . . . . 5  |-  ( ( ( [ y  /  x ] ph  ->  A. x
( x  =  y  ->  ph ) )  /\  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ph ) )  -> 
( [ y  /  x ] ph  <->  A. x
( x  =  y  ->  ph ) ) )
52, 4mpan2 409 . . . 4  |-  ( ( [ y  /  x ] ph  ->  A. x
( x  =  y  ->  ph ) )  -> 
( [ y  /  x ] ph  <->  A. x
( x  =  y  ->  ph ) ) )
65alimi 1360 . . 3  |-  ( A. x ( [ y  /  x ] ph  ->  A. x ( x  =  y  ->  ph )
)  ->  A. x
( [ y  /  x ] ph  <->  A. x
( x  =  y  ->  ph ) ) )
76orim2i 688 . 2  |-  ( ( A. x  x  =  y  \/  A. x
( [ y  /  x ] ph  ->  A. x
( x  =  y  ->  ph ) ) )  ->  ( A. x  x  =  y  \/  A. x ( [ y  /  x ] ph  <->  A. x ( x  =  y  ->  ph ) ) ) )
81, 7ax-mp 7 1  |-  ( A. x  x  =  y  \/  A. x ( [ y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  ph )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    \/ wo 639   A.wal 1257   [wsb 1661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662
This theorem is referenced by: (None)
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