MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2rexbiia Structured version   Unicode version

Theorem 2rexbiia 2871
Description: Inference adding two restricted existential quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004.)
Hypothesis
Ref Expression
2rexbiia.1  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
2rexbiia  |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. x  e.  A  E. y  e.  B  ps )
Distinct variable groups:    x, y    y, A
Allowed substitution hints:    ph( x, y)    ps( x, y)    A( x)    B( x, y)

Proof of Theorem 2rexbiia
StepHypRef Expression
1 2rexbiia.1 . . 3  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( ph  <->  ps )
)
21rexbidva 2861 . 2  |-  ( x  e.  A  ->  ( E. y  e.  B  ph  <->  E. y  e.  B  ps ) )
32rexbiia 2869 1  |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. x  e.  A  E. y  e.  B  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1758   E.wrex 2799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-ral 2803  df-rex 2804
This theorem is referenced by:  cnref1o  11096  mdsymlem8  25965  xlt2addrd  26201  elunirnmbfm  26811
  Copyright terms: Public domain W3C validator