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Theorem r19.21t 2482
Description: Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers (closed theorem version). (Contributed by NM, 1-Mar-2008.)
Assertion
Ref Expression
r19.21t  |-  ( F/ x ph  ->  ( A. x  e.  A  ( ph  ->  ps )  <->  (
ph  ->  A. x  e.  A  ps ) ) )

Proof of Theorem r19.21t
StepHypRef Expression
1 bi2.04 247 . . . 4  |-  ( ( x  e.  A  -> 
( ph  ->  ps )
)  <->  ( ph  ->  ( x  e.  A  ->  ps ) ) )
21albii 1429 . . 3  |-  ( A. x ( x  e.  A  ->  ( ph  ->  ps ) )  <->  A. x
( ph  ->  ( x  e.  A  ->  ps ) ) )
3 19.21t 1544 . . 3  |-  ( F/ x ph  ->  ( A. x ( ph  ->  ( x  e.  A  ->  ps ) )  <->  ( ph  ->  A. x ( x  e.  A  ->  ps ) ) ) )
42, 3syl5bb 191 . 2  |-  ( F/ x ph  ->  ( A. x ( x  e.  A  ->  ( ph  ->  ps ) )  <->  ( ph  ->  A. x ( x  e.  A  ->  ps ) ) ) )
5 df-ral 2396 . 2  |-  ( A. x  e.  A  ( ph  ->  ps )  <->  A. x
( x  e.  A  ->  ( ph  ->  ps ) ) )
6 df-ral 2396 . . 3  |-  ( A. x  e.  A  ps  <->  A. x ( x  e.  A  ->  ps )
)
76imbi2i 225 . 2  |-  ( (
ph  ->  A. x  e.  A  ps )  <->  ( ph  ->  A. x ( x  e.  A  ->  ps )
) )
84, 5, 73bitr4g 222 1  |-  ( F/ x ph  ->  ( A. x  e.  A  ( ph  ->  ps )  <->  (
ph  ->  A. x  e.  A  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1312   F/wnf 1419    e. wcel 1463   A.wral 2391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-4 1470  ax-ial 1497  ax-i5r 1498
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-ral 2396
This theorem is referenced by:  r19.21  2483
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