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Theorem ra5 3005
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 1773. (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 2513 . . . 4  |-  ( A. x  e.  A  ( ph  ->  ps )  <->  A. x
( x  e.  A  ->  ( ph  ->  ps ) ) )
2 bi2.04 352 . . . . 5  |-  ( ( x  e.  A  -> 
( ph  ->  ps )
)  <->  ( ph  ->  ( x  e.  A  ->  ps ) ) )
32albii 1554 . . . 4  |-  ( A. x ( x  e.  A  ->  ( ph  ->  ps ) )  <->  A. x
( ph  ->  ( x  e.  A  ->  ps ) ) )
41, 3bitri 242 . . 3  |-  ( A. x  e.  A  ( ph  ->  ps )  <->  A. x
( ph  ->  ( x  e.  A  ->  ps ) ) )
5 ra5.1 . . . 4  |-  F/ x ph
65stdpc5 1773 . . 3  |-  ( A. x ( ph  ->  ( x  e.  A  ->  ps ) )  ->  ( ph  ->  A. x ( x  e.  A  ->  ps ) ) )
74, 6sylbi 189 . 2  |-  ( A. x  e.  A  ( ph  ->  ps )  -> 
( ph  ->  A. x
( x  e.  A  ->  ps ) ) )
8 df-ral 2513 . 2  |-  ( A. x  e.  A  ps  <->  A. x ( x  e.  A  ->  ps )
)
97, 8syl6ibr 220 1  |-  ( A. x  e.  A  ( ph  ->  ps )  -> 
( ph  ->  A. x  e.  A  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 6   A.wal 1532   F/wnf 1539    e. wcel 1621   A.wral 2509
This theorem is referenced by:  wfr3g  23423  frr3g  23448
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-gen 1536  ax-4 1692
This theorem depends on definitions:  df-bi 179  df-nf 1540  df-ral 2513
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