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Theorem nfsbv 1940
Description: If  z is not free in  ph, it is not free in  [ y  /  x ] ph when  z is distinct from  x and  y. Version of nfsb 1939 requiring more disjoint variables. (Contributed by Wolf Lammen, 7-Feb-2023.) Remove disjoint variable condition on  x ,  y. (Revised by Steven Nguyen, 13-Aug-2023.) Reduce axiom usage. (Revised by Gino Giotto, 25-Aug-2024.)
Hypothesis
Ref Expression
nfsbv.nf  |-  F/ z
ph
Assertion
Ref Expression
nfsbv  |-  F/ z [ y  /  x ] ph
Distinct variable groups:    x, z    y,
z
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem nfsbv
StepHypRef Expression
1 df-sb 1756 . 2  |-  ( [ y  /  x ] ph 
<->  ( ( x  =  y  ->  ph )  /\  E. x ( x  =  y  /\  ph )
) )
2 nfv 1521 . . . 4  |-  F/ z  x  =  y
3 nfsbv.nf . . . 4  |-  F/ z
ph
42, 3nfim 1565 . . 3  |-  F/ z ( x  =  y  ->  ph )
52, 3nfan 1558 . . . 4  |-  F/ z ( x  =  y  /\  ph )
65nfex 1630 . . 3  |-  F/ z E. x ( x  =  y  /\  ph )
74, 6nfan 1558 . 2  |-  F/ z ( ( x  =  y  ->  ph )  /\  E. x ( x  =  y  /\  ph )
)
81, 7nfxfr 1467 1  |-  F/ z [ y  /  x ] ph
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   F/wnf 1453   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-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  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:  sbco2v  1941  cbvabw  2293
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