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Theorem sbi2v 1814
Description: Reverse direction of sbimv 1815. (Contributed by Jim Kingdon, 18-Jan-2018.)
Assertion
Ref Expression
sbi2v  |-  ( ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  ->  [ y  /  x ] ( ph  ->  ps ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem sbi2v
StepHypRef Expression
1 19.38 1607 . . 3  |-  ( ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ps ) )  ->  A. x
( ( x  =  y  /\  ph )  ->  ( x  =  y  ->  ps ) ) )
2 pm3.3 257 . . . . 5  |-  ( ( ( x  =  y  /\  ph )  -> 
( x  =  y  ->  ps ) )  ->  ( x  =  y  ->  ( ph  ->  ( x  =  y  ->  ps ) ) ) )
3 pm2.04 81 . . . . 5  |-  ( (
ph  ->  ( x  =  y  ->  ps )
)  ->  ( x  =  y  ->  ( ph  ->  ps ) ) )
42, 3syli 37 . . . 4  |-  ( ( ( x  =  y  /\  ph )  -> 
( x  =  y  ->  ps ) )  ->  ( x  =  y  ->  ( ph  ->  ps ) ) )
54alimi 1385 . . 3  |-  ( A. x ( ( x  =  y  /\  ph )  ->  ( x  =  y  ->  ps )
)  ->  A. x
( x  =  y  ->  ( ph  ->  ps ) ) )
61, 5syl 14 . 2  |-  ( ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ps ) )  ->  A. x
( x  =  y  ->  ( ph  ->  ps ) ) )
7 sb5 1809 . . 3  |-  ( [ y  /  x ] ph 
<->  E. x ( x  =  y  /\  ph ) )
8 sb6 1808 . . 3  |-  ( [ y  /  x ] ps 
<-> 
A. x ( x  =  y  ->  ps ) )
97, 8imbi12i 237 . 2  |-  ( ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  <->  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ps ) ) )
10 sb6 1808 . 2  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  A. x ( x  =  y  ->  ( ph  ->  ps ) ) )
116, 9, 103imtr4i 199 1  |-  ( ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  ->  [ y  /  x ] ( ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   A.wal 1283   E.wex 1422   [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-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-sb 1687
This theorem is referenced by:  sbimv  1815
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