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

Theorem ralbiia 2424
Description: Inference adding restricted universal quantifier to both sides of an equivalence. (Contributed by NM, 26-Nov-2000.)
Hypothesis
Ref Expression
ralbiia.1  |-  ( x  e.  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
ralbiia  |-  ( A. x  e.  A  ph  <->  A. x  e.  A  ps )

Proof of Theorem ralbiia
StepHypRef Expression
1 ralbiia.1 . . 3  |-  ( x  e.  A  ->  ( ph 
<->  ps ) )
21pm5.74i 179 . 2  |-  ( ( x  e.  A  ->  ph )  <->  ( x  e.  A  ->  ps )
)
32ralbii2 2420 1  |-  ( A. x  e.  A  ph  <->  A. x  e.  A  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    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:  frind  4242  poinxp  4576  soinxp  4577  seinxp  4578  dffun8  5119  funcnv3  5153  fncnv  5157  fnres  5207  fvreseq  5490  isoini2  5686  smores  6155  resixp  6593  finomni  6978  caucvgre  10693
  Copyright terms: Public domain W3C validator