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Theorem sbalyz 1917
Description: Move universal quantifier in and out of substitution. Identical to sbal 1918 except that it has an additional distinct variable constraint on  y and  z. (Contributed by Jim Kingdon, 29-Dec-2017.)
Assertion
Ref Expression
sbalyz  |-  ( [ z  /  y ] A. x ph  <->  A. x [ z  /  y ] ph )
Distinct variable group:    x, y, z
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem sbalyz
StepHypRef Expression
1 nfa1 1475 . . . 4  |-  F/ x A. x ph
21nfsbxy 1860 . . 3  |-  F/ x [ z  /  y ] A. x ph
3 ax-4 1441 . . . 4  |-  ( A. x ph  ->  ph )
43sbimi 1688 . . 3  |-  ( [ z  /  y ] A. x ph  ->  [ z  /  y ]
ph )
52, 4alrimi 1456 . 2  |-  ( [ z  /  y ] A. x ph  ->  A. x [ z  / 
y ] ph )
6 sb6 1808 . . . . 5  |-  ( [ z  /  y ]
ph 
<-> 
A. y ( y  =  z  ->  ph )
)
76albii 1400 . . . 4  |-  ( A. x [ z  /  y ] ph  <->  A. x A. y
( y  =  z  ->  ph ) )
8 alcom 1408 . . . 4  |-  ( A. x A. y ( y  =  z  ->  ph )  <->  A. y A. x ( y  =  z  ->  ph ) )
97, 8bitri 182 . . 3  |-  ( A. x [ z  /  y ] ph  <->  A. y A. x
( y  =  z  ->  ph ) )
10 nfv 1462 . . . . . 6  |-  F/ x  y  =  z
1110stdpc5 1517 . . . . 5  |-  ( A. x ( y  =  z  ->  ph )  -> 
( y  =  z  ->  A. x ph )
)
1211alimi 1385 . . . 4  |-  ( A. y A. x ( y  =  z  ->  ph )  ->  A. y ( y  =  z  ->  A. x ph ) )
13 sb2 1691 . . . 4  |-  ( A. y ( y  =  z  ->  A. x ph )  ->  [ z  /  y ] A. x ph )
1412, 13syl 14 . . 3  |-  ( A. y A. x ( y  =  z  ->  ph )  ->  [ z  /  y ] A. x ph )
159, 14sylbi 119 . 2  |-  ( A. x [ z  /  y ] ph  ->  [ z  /  y ] A. x ph )
165, 15impbii 124 1  |-  ( [ z  /  y ] A. x ph  <->  A. x [ z  /  y ] ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1283   [wsb 1686
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687
This theorem is referenced by:  sbal  1918
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