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Theorem cbvexdh 1843
Description: Deduction used to change bound variables, using implicit substitition, particularly useful in conjunction with dvelim 1935. (Contributed by NM, 2-Jan-2002.) (Proof rewritten by Jim Kingdon, 30-Dec-2017.)
Hypotheses
Ref Expression
cbvexdh.1  |-  ( ph  ->  A. y ph )
cbvexdh.2  |-  ( ph  ->  ( ps  ->  A. y ps ) )
cbvexdh.3  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
Assertion
Ref Expression
cbvexdh  |-  ( ph  ->  ( E. x ps  <->  E. y ch ) )
Distinct variable groups:    ph, x    ch, x
Allowed substitution hints:    ph( y)    ps( x, y)    ch( y)

Proof of Theorem cbvexdh
StepHypRef Expression
1 ax-17 1460 . . 3  |-  ( ph  ->  A. x ph )
2 ax-17 1460 . . . 4  |-  ( ch 
->  A. x ch )
32hbex 1568 . . 3  |-  ( E. y ch  ->  A. x E. y ch )
4 cbvexdh.1 . . . . 5  |-  ( ph  ->  A. y ph )
5 cbvexdh.2 . . . . 5  |-  ( ph  ->  ( ps  ->  A. y ps ) )
6 cbvexdh.3 . . . . . 6  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
7 equcomi 1633 . . . . . . 7  |-  ( y  =  x  ->  x  =  y )
8 bicom1 129 . . . . . . 7  |-  ( ( ps  <->  ch )  ->  ( ch 
<->  ps ) )
97, 8imim12i 58 . . . . . 6  |-  ( ( x  =  y  -> 
( ps  <->  ch )
)  ->  ( y  =  x  ->  ( ch  <->  ps ) ) )
106, 9syl 14 . . . . 5  |-  ( ph  ->  ( y  =  x  ->  ( ch  <->  ps )
) )
114, 5, 10equsexd 1658 . . . 4  |-  ( ph  ->  ( E. y ( y  =  x  /\  ch )  <->  ps ) )
12 simpr 108 . . . . 5  |-  ( ( y  =  x  /\  ch )  ->  ch )
1312eximi 1532 . . . 4  |-  ( E. y ( y  =  x  /\  ch )  ->  E. y ch )
1411, 13syl6bir 162 . . 3  |-  ( ph  ->  ( ps  ->  E. y ch ) )
151, 3, 14exlimdh 1528 . 2  |-  ( ph  ->  ( E. x ps 
->  E. y ch )
)
161, 5eximdh 1543 . . . 4  |-  ( ph  ->  ( E. x ps 
->  E. x A. y ps ) )
17 19.12 1596 . . . 4  |-  ( E. x A. y ps 
->  A. y E. x ps )
1816, 17syl6 33 . . 3  |-  ( ph  ->  ( E. x ps 
->  A. y E. x ps ) )
192a1i 9 . . . . 5  |-  ( ph  ->  ( ch  ->  A. x ch ) )
201, 19, 6equsexd 1658 . . . 4  |-  ( ph  ->  ( E. x ( x  =  y  /\  ps )  <->  ch ) )
21 simpr 108 . . . . 5  |-  ( ( x  =  y  /\  ps )  ->  ps )
2221eximi 1532 . . . 4  |-  ( E. x ( x  =  y  /\  ps )  ->  E. x ps )
2320, 22syl6bir 162 . . 3  |-  ( ph  ->  ( ch  ->  E. x ps ) )
244, 18, 23exlimd2 1527 . 2  |-  ( ph  ->  ( E. y ch 
->  E. x ps )
)
2515, 24impbid 127 1  |-  ( ph  ->  ( E. x ps  <->  E. y ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1283    = wceq 1285   E.wex 1422
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-8 1436  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  cbvexd  1844
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