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Theorem 2ralbida 2744
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 1629 . . . 4  |-  F/ y  x  e.  A
42, 3nfan 1846 . . 3  |-  F/ y ( ph  /\  x  e.  A )
5 2ralbida.3 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B ) )  -> 
( ps  <->  ch )
)
65anassrs 630 . . 3  |-  ( ( ( ph  /\  x  e.  A )  /\  y  e.  B )  ->  ( ps 
<->  ch ) )
74, 6ralbida 2719 . 2  |-  ( (
ph  /\  x  e.  A )  ->  ( A. y  e.  B  ps 
<-> 
A. y  e.  B  ch ) )
81, 7ralbida 2719 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 177    /\ wa 359   F/wnf 1553    e. wcel 1725   A.wral 2705
This theorem is referenced by:  2ralbidva  2745
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-11 1761
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554  df-ral 2710
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