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Theorem nfsbxy 1834
Description: Similar to hbsb 1839 but with an extra distinct variable constraint, on  x and  y. (Contributed by Jim Kingdon, 19-Mar-2018.)
Hypothesis
Ref Expression
nfsbxy.1  |-  F/ z
ph
Assertion
Ref Expression
nfsbxy  |-  F/ z [ y  /  x ] ph
Distinct variable groups:    x, y    y,
z
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem nfsbxy
StepHypRef Expression
1 ax-bndl 1415 . 2  |-  ( A. z  z  =  x  \/  ( A. z  z  =  y  \/  A. x A. z ( x  =  y  ->  A. z  x  =  y )
) )
2 nfs1v 1831 . . . 4  |-  F/ z [ y  /  z ] ph
3 drsb1 1696 . . . . 5  |-  ( A. z  z  =  x  ->  ( [ y  / 
z ] ph  <->  [ y  /  x ] ph )
)
43drnf2 1638 . . . 4  |-  ( A. z  z  =  x  ->  ( F/ z [ y  /  z ]
ph 
<->  F/ z [ y  /  x ] ph ) )
52, 4mpbii 140 . . 3  |-  ( A. z  z  =  x  ->  F/ z [ y  /  x ] ph )
6 a16nf 1762 . . . 4  |-  ( A. z  z  =  y  ->  F/ z [ y  /  x ] ph )
7 df-nf 1366 . . . . . 6  |-  ( F/ z  x  =  y  <->  A. z ( x  =  y  ->  A. z  x  =  y )
)
87albii 1375 . . . . 5  |-  ( A. x F/ z  x  =  y  <->  A. x A. z
( x  =  y  ->  A. z  x  =  y ) )
9 sb5 1783 . . . . . 6  |-  ( [ y  /  x ] ph 
<->  E. x ( x  =  y  /\  ph ) )
10 nfa1 1450 . . . . . . 7  |-  F/ x A. x F/ z  x  =  y
11 sp 1417 . . . . . . . 8  |-  ( A. x F/ z  x  =  y  ->  F/ z  x  =  y )
12 nfsbxy.1 . . . . . . . . 9  |-  F/ z
ph
1312a1i 9 . . . . . . . 8  |-  ( A. x F/ z  x  =  y  ->  F/ z ph )
1411, 13nfand 1476 . . . . . . 7  |-  ( A. x F/ z  x  =  y  ->  F/ z
( x  =  y  /\  ph ) )
1510, 14nfexd 1660 . . . . . 6  |-  ( A. x F/ z  x  =  y  ->  F/ z E. x ( x  =  y  /\  ph )
)
169, 15nfxfrd 1380 . . . . 5  |-  ( A. x F/ z  x  =  y  ->  F/ z [ y  /  x ] ph )
178, 16sylbir 129 . . . 4  |-  ( A. x A. z ( x  =  y  ->  A. z  x  =  y )  ->  F/ z [ y  /  x ] ph )
186, 17jaoi 646 . . 3  |-  ( ( A. z  z  =  y  \/  A. x A. z ( x  =  y  ->  A. z  x  =  y )
)  ->  F/ z [ y  /  x ] ph )
195, 18jaoi 646 . 2  |-  ( ( A. z  z  =  x  \/  ( A. z  z  =  y  \/  A. x A. z
( x  =  y  ->  A. z  x  =  y ) ) )  ->  F/ z [ y  /  x ] ph )
201, 19ax-mp 7 1  |-  F/ z [ y  /  x ] ph
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    \/ wo 639   A.wal 1257   F/wnf 1365   E.wex 1397   [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-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662
This theorem is referenced by:  nfsb  1838  sbalyz  1891  opelopabsb  4025
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