ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sb4or Unicode version

Theorem sb4or 1826
Description: One direction of a simplified definition of substitution when variables are distinct. Similar to sb4 1825 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 1823 . 2  |-  ( A. x  x  =  y  \/  ( E. x ( x  =  y  /\  ph )  ->  A. x
( x  =  y  ->  ph ) ) )
2 nfe1 1489 . . . . . 6  |-  F/ x E. x ( x  =  y  /\  ph )
3 nfa1 1534 . . . . . 6  |-  F/ x A. x ( x  =  y  ->  ph )
42, 3nfim 1565 . . . . 5  |-  F/ x
( E. x ( x  =  y  /\  ph )  ->  A. x
( x  =  y  ->  ph ) )
54nfri 1512 . . . 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 1759 . . . . 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 1462 . . 3  |-  ( ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph )
)  ->  A. x
( [ y  /  x ] ph  ->  A. x
( x  =  y  ->  ph ) ) )
98orim2i 756 . 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 703   A.wal 1346   E.wex 1485   [wsb 1755
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756
This theorem is referenced by:  sb4bor  1828  nfsb2or  1830
  Copyright terms: Public domain W3C validator