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Theorem ralbii2 2467
Description: Inference adding different restricted universal quantifiers to each side of an equivalence. (Contributed by NM, 15-Aug-2005.)
Hypothesis
Ref Expression
ralbii2.1  |-  ( ( x  e.  A  ->  ph )  <->  ( x  e.  B  ->  ps )
)
Assertion
Ref Expression
ralbii2  |-  ( A. x  e.  A  ph  <->  A. x  e.  B  ps )

Proof of Theorem ralbii2
StepHypRef Expression
1 ralbii2.1 . . 3  |-  ( ( x  e.  A  ->  ph )  <->  ( x  e.  B  ->  ps )
)
21albii 1450 . 2  |-  ( A. x ( x  e.  A  ->  ph )  <->  A. x
( x  e.  B  ->  ps ) )
3 df-ral 2440 . 2  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
4 df-ral 2440 . 2  |-  ( A. x  e.  B  ps  <->  A. x ( x  e.  B  ->  ps )
)
52, 3, 43bitr4i 211 1  |-  ( A. x  e.  A  ph  <->  A. x  e.  B  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1333    e. wcel 2128   A.wral 2435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429
This theorem depends on definitions:  df-bi 116  df-ral 2440
This theorem is referenced by:  raleqbii  2469  ralbiia  2471  ralrab  2873  raldifb  3247  raluz2  9490  ralrp  9582  isprm4  11995
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