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Theorem nfsb2or 1734
Description: Bound-variable hypothesis builder for substitution. Similar to hbsb2 1733 but in intuitionistic logic a disjunction is stronger than an implication. (Contributed by Jim Kingdon, 2-Feb-2018.)
Assertion
Ref Expression
nfsb2or  |-  ( A. x  x  =  y  \/  F/ x [ y  /  x ] ph )

Proof of Theorem nfsb2or
StepHypRef Expression
1 sb4or 1730 . 2  |-  ( A. x  x  =  y  \/  A. x ( [ y  /  x ] ph  ->  A. x ( x  =  y  ->  ph )
) )
2 sb2 1666 . . . . . . 7  |-  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ph )
32a5i 1451 . . . . . 6  |-  ( A. x ( x  =  y  ->  ph )  ->  A. x [ y  /  x ] ph )
43imim2i 12 . . . . 5  |-  ( ( [ y  /  x ] ph  ->  A. x
( x  =  y  ->  ph ) )  -> 
( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph ) )
54alimi 1360 . . . 4  |-  ( A. x ( [ y  /  x ] ph  ->  A. x ( x  =  y  ->  ph )
)  ->  A. x
( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph ) )
6 df-nf 1366 . . . 4  |-  ( F/ x [ y  /  x ] ph  <->  A. x
( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph ) )
75, 6sylibr 141 . . 3  |-  ( A. x ( [ y  /  x ] ph  ->  A. x ( x  =  y  ->  ph )
)  ->  F/ x [ y  /  x ] ph )
87orim2i 688 . 2  |-  ( ( A. x  x  =  y  \/  A. x
( [ y  /  x ] ph  ->  A. x
( x  =  y  ->  ph ) ) )  ->  ( A. x  x  =  y  \/  F/ x [ y  /  x ] ph ) )
91, 8ax-mp 7 1  |-  ( A. x  x  =  y  \/  F/ x [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 639   A.wal 1257   F/wnf 1365   [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:  sbequi  1736
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