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Theorem equsexd 1664
Description: Deduction form of equsex 1663. (Contributed by Jim Kingdon, 29-Dec-2017.)
Hypotheses
Ref Expression
equsexd.1  |-  ( ph  ->  A. x ph )
equsexd.2  |-  ( ph  ->  ( ch  ->  A. x ch ) )
equsexd.3  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
Assertion
Ref Expression
equsexd  |-  ( ph  ->  ( E. x ( x  =  y  /\  ps )  <->  ch ) )

Proof of Theorem equsexd
StepHypRef Expression
1 equsexd.1 . . 3  |-  ( ph  ->  A. x ph )
2 equsexd.2 . . 3  |-  ( ph  ->  ( ch  ->  A. x ch ) )
3 equsexd.3 . . . 4  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
4 bi1 116 . . . . 5  |-  ( ( ps  <->  ch )  ->  ( ps  ->  ch ) )
54imim2i 12 . . . 4  |-  ( ( x  =  y  -> 
( ps  <->  ch )
)  ->  ( x  =  y  ->  ( ps 
->  ch ) ) )
6 pm3.31 258 . . . 4  |-  ( ( x  =  y  -> 
( ps  ->  ch ) )  ->  (
( x  =  y  /\  ps )  ->  ch ) )
73, 5, 63syl 17 . . 3  |-  ( ph  ->  ( ( x  =  y  /\  ps )  ->  ch ) )
81, 2, 7exlimd2 1531 . 2  |-  ( ph  ->  ( E. x ( x  =  y  /\  ps )  ->  ch )
)
9 a9e 1631 . . . 4  |-  E. x  x  =  y
101a1i 9 . . . . . . . . 9  |-  ( ph  ->  ( ph  ->  A. x ph ) )
1110, 2jca 300 . . . . . . . 8  |-  ( ph  ->  ( ( ph  ->  A. x ph )  /\  ( ch  ->  A. x ch ) ) )
12 prth 336 . . . . . . . 8  |-  ( ( ( ph  ->  A. x ph )  /\  ( ch  ->  A. x ch )
)  ->  ( ( ph  /\  ch )  -> 
( A. x ph  /\ 
A. x ch )
) )
1311, 12syl 14 . . . . . . 7  |-  ( ph  ->  ( ( ph  /\  ch )  ->  ( A. x ph  /\  A. x ch ) ) )
14 19.26 1415 . . . . . . 7  |-  ( A. x ( ph  /\  ch )  <->  ( A. x ph  /\  A. x ch ) )
1513, 14syl6ibr 160 . . . . . 6  |-  ( ph  ->  ( ( ph  /\  ch )  ->  A. x
( ph  /\  ch )
) )
1615anabsi5 546 . . . . 5  |-  ( (
ph  /\  ch )  ->  A. x ( ph  /\ 
ch ) )
17 idd 21 . . . . . . . 8  |-  ( ch 
->  ( x  =  y  ->  x  =  y ) )
1817a1i 9 . . . . . . 7  |-  ( ph  ->  ( ch  ->  (
x  =  y  ->  x  =  y )
) )
1918imp 122 . . . . . 6  |-  ( (
ph  /\  ch )  ->  ( x  =  y  ->  x  =  y ) )
20 bi2 128 . . . . . . . . 9  |-  ( ( ps  <->  ch )  ->  ( ch  ->  ps ) )
2120imim2i 12 . . . . . . . 8  |-  ( ( x  =  y  -> 
( ps  <->  ch )
)  ->  ( x  =  y  ->  ( ch 
->  ps ) ) )
22 pm2.04 81 . . . . . . . 8  |-  ( ( x  =  y  -> 
( ch  ->  ps ) )  ->  ( ch  ->  ( x  =  y  ->  ps )
) )
233, 21, 223syl 17 . . . . . . 7  |-  ( ph  ->  ( ch  ->  (
x  =  y  ->  ps ) ) )
2423imp 122 . . . . . 6  |-  ( (
ph  /\  ch )  ->  ( x  =  y  ->  ps ) )
2519, 24jcad 301 . . . . 5  |-  ( (
ph  /\  ch )  ->  ( x  =  y  ->  ( x  =  y  /\  ps )
) )
2616, 25eximdh 1547 . . . 4  |-  ( (
ph  /\  ch )  ->  ( E. x  x  =  y  ->  E. x
( x  =  y  /\  ps ) ) )
279, 26mpi 15 . . 3  |-  ( (
ph  /\  ch )  ->  E. x ( x  =  y  /\  ps ) )
2827ex 113 . 2  |-  ( ph  ->  ( ch  ->  E. x
( x  =  y  /\  ps ) ) )
298, 28impbid 127 1  |-  ( ph  ->  ( E. x ( x  =  y  /\  ps )  <->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1287    = wceq 1289   E.wex 1426
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 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-i9 1468  ax-ial 1472
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  cbvexdh  1849
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