ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ralbii2 Unicode version

Theorem ralbii2 2420
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 1429 . 2  |-  ( A. x ( x  e.  A  ->  ph )  <->  A. x
( x  e.  B  ->  ps ) )
3 df-ral 2396 . 2  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
4 df-ral 2396 . 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 1312    e. wcel 1463   A.wral 2391
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 1406  ax-gen 1408
This theorem depends on definitions:  df-bi 116  df-ral 2396
This theorem is referenced by:  raleqbii  2422  ralbiia  2424  ralrab  2816  raldifb  3184  raluz2  9323  ralrp  9411  isprm4  11696
  Copyright terms: Public domain W3C validator