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Theorem gencbvex2 3060
Description: Restatement of gencbvex 3059 with weaker hypotheses. (Contributed by Jeff Hankins, 6-Dec-2006.)
Hypotheses
Ref Expression
gencbvex2.1  |-  A  e. 
_V
gencbvex2.2  |-  ( A  =  y  ->  ( ph 
<->  ps ) )
gencbvex2.3  |-  ( A  =  y  ->  ( ch 
<->  th ) )
gencbvex2.4  |-  ( th 
->  E. x ( ch 
/\  A  =  y ) )
Assertion
Ref Expression
gencbvex2  |-  ( E. x ( ch  /\  ph )  <->  E. y ( th 
/\  ps ) )
Distinct variable groups:    ps, x    ph, y    th, x    ch, y    y, A
Allowed substitution hints:    ph( x)    ps( y)    ch( x)    th( y)    A( x)

Proof of Theorem gencbvex2
StepHypRef Expression
1 gencbvex2.1 . 2  |-  A  e. 
_V
2 gencbvex2.2 . 2  |-  ( A  =  y  ->  ( ph 
<->  ps ) )
3 gencbvex2.3 . 2  |-  ( A  =  y  ->  ( ch 
<->  th ) )
4 gencbvex2.4 . . 3  |-  ( th 
->  E. x ( ch 
/\  A  =  y ) )
53biimpac 493 . . . 4  |-  ( ( ch  /\  A  =  y )  ->  th )
65exlimiv 1779 . . 3  |-  ( E. x ( ch  /\  A  =  y )  ->  th )
74, 6impbii 192 . 2  |-  ( th  <->  E. x ( ch  /\  A  =  y )
)
81, 2, 3, 7gencbvex 3059 1  |-  ( E. x ( ch  /\  ph )  <->  E. y ( th 
/\  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1447   E.wex 1666    e. wcel 1890   _Vcvv 3012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-10 1918  ax-11 1923  ax-12 1936  ax-ext 2431
This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1450  df-ex 1667  df-nf 1671  df-sb 1801  df-clab 2438  df-cleq 2444  df-clel 2447  df-v 3014
This theorem is referenced by: (None)
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