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Theorem sb4or 1787
Description: One direction of a simplified definition of substitution when variables are distinct. Similar to sb4 1786 but stronger in intuitionistic logic. (Contributed by Jim Kingdon, 2-Feb-2018.)
Assertion
Ref Expression
sb4or  |-  ( A. x  x  =  y  \/  A. x ( [ y  /  x ] ph  ->  A. x ( x  =  y  ->  ph )
) )

Proof of Theorem sb4or
StepHypRef Expression
1 equs5or 1784 . 2  |-  ( A. x  x  =  y  \/  ( E. x ( x  =  y  /\  ph )  ->  A. x
( x  =  y  ->  ph ) ) )
2 nfe1 1455 . . . . . 6  |-  F/ x E. x ( x  =  y  /\  ph )
3 nfa1 1504 . . . . . 6  |-  F/ x A. x ( x  =  y  ->  ph )
42, 3nfim 1534 . . . . 5  |-  F/ x
( E. x ( x  =  y  /\  ph )  ->  A. x
( x  =  y  ->  ph ) )
54nfri 1482 . . . 4  |-  ( ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph )
)  ->  A. x
( E. x ( x  =  y  /\  ph )  ->  A. x
( x  =  y  ->  ph ) ) )
6 sb1 1722 . . . . 5  |-  ( [ y  /  x ] ph  ->  E. x ( x  =  y  /\  ph ) )
76imim1i 60 . . . 4  |-  ( ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph )
)  ->  ( [
y  /  x ] ph  ->  A. x ( x  =  y  ->  ph )
) )
85, 7alrimih 1428 . . 3  |-  ( ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph )
)  ->  A. x
( [ y  /  x ] ph  ->  A. x
( x  =  y  ->  ph ) ) )
98orim2i 733 . 2  |-  ( ( A. x  x  =  y  \/  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph )
) )  ->  ( A. x  x  =  y  \/  A. x
( [ y  /  x ] ph  ->  A. x
( x  =  y  ->  ph ) ) ) )
101, 9ax-mp 5 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 103    \/ wo 680   A.wal 1312   E.wex 1451   [wsb 1718
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-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719
This theorem is referenced by:  sb4bor  1789  nfsb2or  1791
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