MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rr19.3v Unicode version

Theorem rr19.3v 2922
Description: Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. We don't need the non-empty class condition of r19.3rzv 3560 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 228 . . . 4  |-  ( y  =  x  ->  ( ph 
<-> 
ph ) )
21rspcv 2893 . . 3  |-  ( x  e.  A  ->  ( A. y  e.  A  ph 
->  ph ) )
32ralimia 2629 . 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 2638 . . 3  |-  ( ph  ->  A. y  e.  A  ph )
65ralimi 2631 . 2  |-  ( A. x  e.  A  ph  ->  A. x  e.  A  A. y  e.  A  ph )
73, 6impbii 180 1  |-  ( A. x  e.  A  A. y  e.  A  ph  <->  A. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1632    e. wcel 1696   A.wral 2556
This theorem is referenced by:  ispos2  14098
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-v 2803
  Copyright terms: Public domain W3C validator