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Theorem sbalyz 1972
Description: Move universal quantifier in and out of substitution. Identical to sbal 1973 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 1521 . . . 4  |-  F/ x A. x ph
21nfsbxy 1913 . . 3  |-  F/ x [ z  /  y ] A. x ph
3 ax-4 1487 . . . 4  |-  ( A. x ph  ->  ph )
43sbimi 1737 . . 3  |-  ( [ z  /  y ] A. x ph  ->  [ z  /  y ]
ph )
52, 4alrimi 1502 . 2  |-  ( [ z  /  y ] A. x ph  ->  A. x [ z  / 
y ] ph )
6 sb6 1858 . . . . 5  |-  ( [ z  /  y ]
ph 
<-> 
A. y ( y  =  z  ->  ph )
)
76albii 1446 . . . 4  |-  ( A. x [ z  /  y ] ph  <->  A. x A. y
( y  =  z  ->  ph ) )
8 alcom 1454 . . . 4  |-  ( A. x A. y ( y  =  z  ->  ph )  <->  A. y A. x ( y  =  z  ->  ph ) )
97, 8bitri 183 . . 3  |-  ( A. x [ z  /  y ] ph  <->  A. y A. x
( y  =  z  ->  ph ) )
10 nfv 1508 . . . . . 6  |-  F/ x  y  =  z
1110stdpc5 1563 . . . . 5  |-  ( A. x ( y  =  z  ->  ph )  -> 
( y  =  z  ->  A. x ph )
)
1211alimi 1431 . . . 4  |-  ( A. y A. x ( y  =  z  ->  ph )  ->  A. y ( y  =  z  ->  A. x ph ) )
13 sb2 1740 . . . 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 120 . 2  |-  ( A. x [ z  /  y ] ph  ->  [ z  /  y ] A. x ph )
165, 15impbii 125 1  |-  ( [ z  /  y ] A. x ph  <->  A. x [ z  /  y ] ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1329   [wsb 1735
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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736
This theorem is referenced by:  sbal  1973
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