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Theorem rr19.3v 2846
Description: Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. We don't need the non-empty class condition of r19.3rzv 3453 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 230 . . . 4  |-  ( y  =  x  ->  ( ph 
<-> 
ph ) )
21rcla4v 2817 . . 3  |-  ( x  e.  A  ->  ( A. y  e.  A  ph 
->  ph ) )
32ralimia 2578 . 2  |-  ( A. x  e.  A  A. y  e.  A  ph  ->  A. x  e.  A  ph )
4 ax-1 7 . . . 4  |-  ( ph  ->  ( y  e.  A  ->  ph ) )
54ralrimiv 2587 . . 3  |-  ( ph  ->  A. y  e.  A  ph )
65ralimi 2580 . 2  |-  ( A. x  e.  A  ph  ->  A. x  e.  A  A. y  e.  A  ph )
73, 6impbii 182 1  |-  ( A. x  e.  A  A. y  e.  A  ph  <->  A. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1619    e. wcel 1621   A.wral 2509
This theorem is referenced by:  ispos2  13926
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ral 2513  df-v 2729
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