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Theorem hbsbv 1860
Description: This is a version of hbsb 1866 with an extra distinct variable constraint, on  z and  x. (Contributed by Jim Kingdon, 25-Dec-2017.)
Hypothesis
Ref Expression
hbsbv.1  |-  ( ph  ->  A. z ph )
Assertion
Ref Expression
hbsbv  |-  ( [ y  /  x ] ph  ->  A. z [ y  /  x ] ph )
Distinct variable groups:    x, z    y,
z
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem hbsbv
StepHypRef Expression
1 df-sb 1688 . 2  |-  ( [ y  /  x ] ph 
<->  ( ( x  =  y  ->  ph )  /\  E. x ( x  =  y  /\  ph )
) )
2 ax-17 1460 . . . 4  |-  ( x  =  y  ->  A. z  x  =  y )
3 hbsbv.1 . . . 4  |-  ( ph  ->  A. z ph )
42, 3hbim 1478 . . 3  |-  ( ( x  =  y  ->  ph )  ->  A. z
( x  =  y  ->  ph ) )
52, 3hban 1480 . . . 4  |-  ( ( x  =  y  /\  ph )  ->  A. z
( x  =  y  /\  ph ) )
65hbex 1568 . . 3  |-  ( E. x ( x  =  y  /\  ph )  ->  A. z E. x
( x  =  y  /\  ph ) )
74, 6hban 1480 . 2  |-  ( ( ( x  =  y  ->  ph )  /\  E. x ( x  =  y  /\  ph )
)  ->  A. z
( ( x  =  y  ->  ph )  /\  E. x ( x  =  y  /\  ph )
) )
81, 7hbxfrbi 1402 1  |-  ( [ y  /  x ] ph  ->  A. z [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   A.wal 1283   E.wex 1422   [wsb 1687
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-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-sb 1688
This theorem is referenced by:  sbco2vlem  1863  2sb5rf  1908  2sb6rf  1909
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