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Theorem rexraleqim 3197
Description: Statement following from existence and generalization with equality. (Contributed by AV, 9-Feb-2019.)
Hypotheses
Ref Expression
rexraleqim.1  |-  ( x  =  z  ->  ( ps 
<-> 
ph ) )
rexraleqim.2  |-  ( z  =  Y  ->  ( ph 
<->  th ) )
Assertion
Ref Expression
rexraleqim  |-  ( ( E. z  e.  A  ph 
/\  A. x  e.  A  ( ps  ->  x  =  Y ) )  ->  th )
Distinct variable groups:    x, A, z    x, Y, z    ph, x    ps, z    th, z
Allowed substitution hints:    ph( z)    ps( x)    th( x)

Proof of Theorem rexraleqim
StepHypRef Expression
1 rexraleqim.1 . . . . . . 7  |-  ( x  =  z  ->  ( ps 
<-> 
ph ) )
2 eqeq1 2426 . . . . . . 7  |-  ( x  =  z  ->  (
x  =  Y  <->  z  =  Y ) )
31, 2imbi12d 321 . . . . . 6  |-  ( x  =  z  ->  (
( ps  ->  x  =  Y )  <->  ( ph  ->  z  =  Y ) ) )
43rspcva 3180 . . . . 5  |-  ( ( z  e.  A  /\  A. x  e.  A  ( ps  ->  x  =  Y ) )  -> 
( ph  ->  z  =  Y ) )
5 rexraleqim.2 . . . . . 6  |-  ( z  =  Y  ->  ( ph 
<->  th ) )
65biimpd 210 . . . . 5  |-  ( z  =  Y  ->  ( ph  ->  th ) )
74, 6syli 38 . . . 4  |-  ( ( z  e.  A  /\  A. x  e.  A  ( ps  ->  x  =  Y ) )  -> 
( ph  ->  th )
)
87impancom 441 . . 3  |-  ( ( z  e.  A  /\  ph )  ->  ( A. x  e.  A  ( ps  ->  x  =  Y )  ->  th )
)
98rexlimiva 2910 . 2  |-  ( E. z  e.  A  ph  ->  ( A. x  e.  A  ( ps  ->  x  =  Y )  ->  th ) )
109imp 430 1  |-  ( ( E. z  e.  A  ph 
/\  A. x  e.  A  ( ps  ->  x  =  Y ) )  ->  th )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872   A.wral 2771   E.wrex 2772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ral 2776  df-rex 2777  df-v 3082
This theorem is referenced by:  cramerlem3  19713
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