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Theorem gencbvex2 2666
Description: Restatement of gencbvex 2665 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 292 . . . 4  |-  ( ( ch  /\  A  =  y )  ->  th )
65exlimiv 1534 . . 3  |-  ( E. x ( ch  /\  A  =  y )  ->  th )
74, 6impbii 124 . 2  |-  ( th  <->  E. x ( ch  /\  A  =  y )
)
81, 2, 3, 7gencbvex 2665 1  |-  ( E. x ( ch  /\  ph )  <->  E. y ( th 
/\  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289   E.wex 1426    e. wcel 1438   _Vcvv 2619
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-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621
This theorem is referenced by: (None)
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