MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reubiia Structured version   Unicode version

Theorem reubiia 2895
Description: Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 14-Nov-2004.)
Hypothesis
Ref Expression
reubiia.1  |-  ( x  e.  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
reubiia  |-  ( E! x  e.  A  ph  <->  E! x  e.  A  ps )

Proof of Theorem reubiia
StepHypRef Expression
1 reubiia.1 . . . 4  |-  ( x  e.  A  ->  ( ph 
<->  ps ) )
21pm5.32i 620 . . 3  |-  ( ( x  e.  A  /\  ph )  <->  ( x  e.  A  /\  ps )
)
32eubii 2292 . 2  |-  ( E! x ( x  e.  A  /\  ph )  <->  E! x ( x  e.  A  /\  ps )
)
4 df-reu 2714 . 2  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
5 df-reu 2714 . 2  |-  ( E! x  e.  A  ps  <->  E! x ( x  e.  A  /\  ps )
)
63, 4, 53bitr4i 270 1  |-  ( E! x  e.  A  ph  <->  E! x  e.  A  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    e. wcel 1726   E!weu 2283   E!wreu 2709
This theorem is referenced by:  reubii  2896  riotaxfrd  6583
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-11 1762
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-eu 2287  df-reu 2714
  Copyright terms: Public domain W3C validator