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Theorem bj-sbimedh 10718
Description: A strengthening of sbiedh 1711 (same proof). (Contributed by BJ, 16-Dec-2019.)
Hypotheses
Ref Expression
bj-sbimedh.1  |-  ( ph  ->  A. x ph )
bj-sbimedh.2  |-  ( ph  ->  ( ch  ->  A. x ch ) )
bj-sbimedh.3  |-  ( ph  ->  ( x  =  y  ->  ( ps  ->  ch ) ) )
Assertion
Ref Expression
bj-sbimedh  |-  ( ph  ->  ( [ y  /  x ] ps  ->  ch ) )

Proof of Theorem bj-sbimedh
StepHypRef Expression
1 sb1 1690 . . 3  |-  ( [ y  /  x ] ps  ->  E. x ( x  =  y  /\  ps ) )
2 bj-sbimedh.1 . . . 4  |-  ( ph  ->  A. x ph )
3 bj-sbimedh.3 . . . . 5  |-  ( ph  ->  ( x  =  y  ->  ( ps  ->  ch ) ) )
43impd 251 . . . 4  |-  ( ph  ->  ( ( x  =  y  /\  ps )  ->  ch ) )
52, 4eximdh 1543 . . 3  |-  ( ph  ->  ( E. x ( x  =  y  /\  ps )  ->  E. x ch ) )
61, 5syl5 32 . 2  |-  ( ph  ->  ( [ y  /  x ] ps  ->  E. x ch ) )
7 bj-sbimedh.2 . . 3  |-  ( ph  ->  ( ch  ->  A. x ch ) )
82, 719.9hd 1593 . 2  |-  ( ph  ->  ( E. x ch 
->  ch ) )
96, 8syld 44 1  |-  ( ph  ->  ( [ y  /  x ] ps  ->  ch ) )
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-4 1441  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-sb 1687
This theorem is referenced by:  bj-sbimeh  10719
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