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Theorem rr19.3v 3208
Description: Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. We don't need the nonempty class condition of r19.3rzv 3882 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 237 . . . 4  |-  ( y  =  x  ->  ( ph 
<-> 
ph ) )
21rspcv 3175 . . 3  |-  ( x  e.  A  ->  ( A. y  e.  A  ph 
->  ph ) )
32ralimia 2817 . 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 2828 . . 3  |-  ( ph  ->  A. y  e.  A  ph )
65ralimi 2819 . 2  |-  ( A. x  e.  A  ph  ->  A. x  e.  A  A. y  e.  A  ph )
73, 6impbii 188 1  |-  ( A. x  e.  A  A. y  e.  A  ph  <->  A. x  e.  A  ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    e. wcel 1758   A.wral 2799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-v 3080
This theorem is referenced by:  ispos2  15238
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