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

Theorem 2ralbida 2908
Description: Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 24-Feb-2004.)
Hypotheses
Ref Expression
2ralbida.1  |-  F/ x ph
2ralbida.2  |-  F/ y
ph
2ralbida.3  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B ) )  -> 
( ps  <->  ch )
)
Assertion
Ref Expression
2ralbida  |-  ( ph  ->  ( A. x  e.  A  A. y  e.  B  ps  <->  A. x  e.  A  A. y  e.  B  ch )
)
Distinct variable groups:    x, y    y, A
Allowed substitution hints:    ph( x, y)    ps( x, y)    ch( x, y)    A( x)    B( x, y)

Proof of Theorem 2ralbida
StepHypRef Expression
1 2ralbida.1 . 2  |-  F/ x ph
2 2ralbida.2 . . . 4  |-  F/ y
ph
3 nfv 1683 . . . 4  |-  F/ y  x  e.  A
42, 3nfan 1875 . . 3  |-  F/ y ( ph  /\  x  e.  A )
5 2ralbida.3 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B ) )  -> 
( ps  <->  ch )
)
65anassrs 648 . . 3  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  B )  ->  ( ps 
<->  ch ) )
74, 6ralbida 2900 . 2  |-  ( (
ph  /\  x  e.  A )  ->  ( A. y  e.  B  ps 
<-> 
A. y  e.  B  ch ) )
81, 7ralbida 2900 1  |-  ( ph  ->  ( A. x  e.  A  A. y  e.  B  ps  <->  A. x  e.  A  A. y  e.  B  ch )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   F/wnf 1599    e. wcel 1767   A.wral 2817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-12 1803
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1597  df-nf 1600  df-ral 2822
This theorem is referenced by:  2ralbidvaOLD  2910
  Copyright terms: Public domain W3C validator