MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ra5 Unicode version

Theorem ra5 3151
Description: Restricted quantifier version of Axiom 5 of [Mendelson] p. 69. This is an axiom of a predicate calculus for a restricted domain. Compare the unrestricted stdpc5 1799. (Contributed by NM, 16-Jan-2004.)
Hypothesis
Ref Expression
ra5.1  |-  F/ x ph
Assertion
Ref Expression
ra5  |-  ( A. x  e.  A  ( ph  ->  ps )  -> 
( ph  ->  A. x  e.  A  ps )
)

Proof of Theorem ra5
StepHypRef Expression
1 df-ral 2624 . . . 4  |-  ( A. x  e.  A  ( ph  ->  ps )  <->  A. x
( x  e.  A  ->  ( ph  ->  ps ) ) )
2 bi2.04 350 . . . . 5  |-  ( ( x  e.  A  -> 
( ph  ->  ps )
)  <->  ( ph  ->  ( x  e.  A  ->  ps ) ) )
32albii 1566 . . . 4  |-  ( A. x ( x  e.  A  ->  ( ph  ->  ps ) )  <->  A. x
( ph  ->  ( x  e.  A  ->  ps ) ) )
41, 3bitri 240 . . 3  |-  ( A. x  e.  A  ( ph  ->  ps )  <->  A. x
( ph  ->  ( x  e.  A  ->  ps ) ) )
5 ra5.1 . . . 4  |-  F/ x ph
65stdpc5 1799 . . 3  |-  ( A. x ( ph  ->  ( x  e.  A  ->  ps ) )  ->  ( ph  ->  A. x ( x  e.  A  ->  ps ) ) )
74, 6sylbi 187 . 2  |-  ( A. x  e.  A  ( ph  ->  ps )  -> 
( ph  ->  A. x
( x  e.  A  ->  ps ) ) )
8 df-ral 2624 . 2  |-  ( A. x  e.  A  ps  <->  A. x ( x  e.  A  ->  ps )
)
97, 8syl6ibr 218 1  |-  ( A. x  e.  A  ( ph  ->  ps )  -> 
( ph  ->  A. x  e.  A  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1540   F/wnf 1544    e. wcel 1710   A.wral 2619
This theorem is referenced by:  wfr3g  24813  frr3g  24838
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746
This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545  df-ral 2624
  Copyright terms: Public domain W3C validator