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Theorem 2ralbida 2582
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 1605 . . . 4  |-  F/ y  x  e.  A
42, 3nfan 1771 . . 3  |-  F/ y ( ph  /\  x  e.  A )
5 2ralbida.3 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B ) )  -> 
( ps  <->  ch )
)
65anassrs 629 . . 3  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  B )  ->  ( ps 
<->  ch ) )
74, 6ralbida 2557 . 2  |-  ( (
ph  /\  x  e.  A )  ->  ( A. y  e.  B  ps 
<-> 
A. y  e.  B  ch ) )
81, 7ralbida 2557 1  |-  ( ph  ->  ( A. x  e.  A  A. y  e.  B  ps  <->  A. x  e.  A  A. y  e.  B  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   F/wnf 1531    e. wcel 1684   A.wral 2543
This theorem is referenced by:  2ralbidva  2583
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-nf 1532  df-ral 2548
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