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Theorem rr19.28v 2792
Description: Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 29-Oct-2012.)
Assertion
Ref Expression
rr19.28v  |-  ( A. x  e.  A  A. y  e.  A  ( ph  /\  ps )  <->  A. x  e.  A  ( ph  /\ 
A. y  e.  A  ps ) )
Distinct variable groups:    y, A    x, y    ph, y
Allowed substitution hints:    ph( x)    ps( x, y)    A( x)

Proof of Theorem rr19.28v
StepHypRef Expression
1 simpl 108 . . . . . 6  |-  ( (
ph  /\  ps )  ->  ph )
21ralimi 2467 . . . . 5  |-  ( A. y  e.  A  ( ph  /\  ps )  ->  A. y  e.  A  ph )
3 biidd 171 . . . . . 6  |-  ( y  =  x  ->  ( ph 
<-> 
ph ) )
43rspcv 2754 . . . . 5  |-  ( x  e.  A  ->  ( A. y  e.  A  ph 
->  ph ) )
52, 4syl5 32 . . . 4  |-  ( x  e.  A  ->  ( A. y  e.  A  ( ph  /\  ps )  ->  ph ) )
6 simpr 109 . . . . . 6  |-  ( (
ph  /\  ps )  ->  ps )
76ralimi 2467 . . . . 5  |-  ( A. y  e.  A  ( ph  /\  ps )  ->  A. y  e.  A  ps )
87a1i 9 . . . 4  |-  ( x  e.  A  ->  ( A. y  e.  A  ( ph  /\  ps )  ->  A. y  e.  A  ps ) )
95, 8jcad 303 . . 3  |-  ( x  e.  A  ->  ( A. y  e.  A  ( ph  /\  ps )  ->  ( ph  /\  A. y  e.  A  ps ) ) )
109ralimia 2465 . 2  |-  ( A. x  e.  A  A. y  e.  A  ( ph  /\  ps )  ->  A. x  e.  A  ( ph  /\  A. y  e.  A  ps )
)
11 r19.28av 2540 . . 3  |-  ( (
ph  /\  A. y  e.  A  ps )  ->  A. y  e.  A  ( ph  /\  ps )
)
1211ralimi 2467 . 2  |-  ( A. x  e.  A  ( ph  /\  A. y  e.  A  ps )  ->  A. x  e.  A  A. y  e.  A  ( ph  /\  ps )
)
1310, 12impbii 125 1  |-  ( A. x  e.  A  A. y  e.  A  ( ph  /\  ps )  <->  A. x  e.  A  ( ph  /\ 
A. y  e.  A  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    e. wcel 1461   A.wral 2388
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-v 2657
This theorem is referenced by: (None)
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