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Theorem sb4or 1761
Description: One direction of a simplified definition of substitution when variables are distinct. Similar to sb4 1760 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 1758 . 2  |-  ( A. x  x  =  y  \/  ( E. x ( x  =  y  /\  ph )  ->  A. x
( x  =  y  ->  ph ) ) )
2 nfe1 1430 . . . . . 6  |-  F/ x E. x ( x  =  y  /\  ph )
3 nfa1 1479 . . . . . 6  |-  F/ x A. x ( x  =  y  ->  ph )
42, 3nfim 1509 . . . . 5  |-  F/ x
( E. x ( x  =  y  /\  ph )  ->  A. x
( x  =  y  ->  ph ) )
54nfri 1457 . . . 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 1696 . . . . 5  |-  ( [ y  /  x ] ph  ->  E. x ( x  =  y  /\  ph ) )
76imim1i 59 . . . 4  |-  ( ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph )
)  ->  ( [
y  /  x ] ph  ->  A. x ( x  =  y  ->  ph )
) )
85, 7alrimih 1403 . . 3  |-  ( ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph )
)  ->  A. x
( [ y  /  x ] ph  ->  A. x
( x  =  y  ->  ph ) ) )
98orim2i 713 . 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 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 102    \/ wo 664   A.wal 1287   E.wex 1426   [wsb 1692
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693
This theorem is referenced by:  sb4bor  1763  nfsb2or  1765
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