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Theorem rr19.3v 3069
Description: Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. We don't need the non-empty class condition of r19.3rzv 3713 when there is an outer quantifier. (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 229 . . . 4  |-  ( y  =  x  ->  ( ph 
<-> 
ph ) )
21rspcv 3040 . . 3  |-  ( x  e.  A  ->  ( A. y  e.  A  ph 
->  ph ) )
32ralimia 2771 . 2  |-  ( A. x  e.  A  A. y  e.  A  ph  ->  A. x  e.  A  ph )
4 ax-1 5 . . . 4  |-  ( ph  ->  ( y  e.  A  ->  ph ) )
54ralrimiv 2780 . . 3  |-  ( ph  ->  A. y  e.  A  ph )
65ralimi 2773 . 2  |-  ( A. x  e.  A  ph  ->  A. x  e.  A  A. y  e.  A  ph )
73, 6impbii 181 1  |-  ( A. x  e.  A  A. y  e.  A  ph  <->  A. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    e. wcel 1725   A.wral 2697
This theorem is referenced by:  ispos2  14393
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-v 2950
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