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Theorem rr19.3v 2876
Description: Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. (Contributed by NM, 25-Oct-2012.)
Assertion
Ref Expression
rr19.3v  |-  ( A. x  e.  A  A. y  e.  A  ph  <->  A. x  e.  A  ph )
Distinct variable groups:    y, A    x, y    ph, y
Allowed substitution hints:    ph( x)    A( x)

Proof of Theorem rr19.3v
StepHypRef Expression
1 biidd 172 . . . 4  |-  ( y  =  x  ->  ( ph 
<-> 
ph ) )
21rspcv 2837 . . 3  |-  ( x  e.  A  ->  ( A. y  e.  A  ph 
->  ph ) )
32ralimia 2538 . 2  |-  ( A. x  e.  A  A. y  e.  A  ph  ->  A. x  e.  A  ph )
4 ax-1 6 . . . 4  |-  ( ph  ->  ( y  e.  A  ->  ph ) )
54ralrimiv 2549 . . 3  |-  ( ph  ->  A. y  e.  A  ph )
65ralimi 2540 . 2  |-  ( A. x  e.  A  ph  ->  A. x  e.  A  A. y  e.  A  ph )
73, 6impbii 126 1  |-  ( A. x  e.  A  A. y  e.  A  ph  <->  A. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 105    e. wcel 2148   A.wral 2455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739
This theorem is referenced by: (None)
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