MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rr19.28v Structured version   Unicode version

Theorem rr19.28v 3196
Description: Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. We don't need the nonempty class condition of r19.28zv 3870 when there is an outer quantifier. (Contributed by NM, 29-Oct-2012.)
Assertion
Ref Expression
rr19.28v  |-  ( A. x  e.  A  A. y  e.  A  ( ph  /\  ps )  <->  A. x  e.  A  ( ph  /\ 
A. y  e.  A  ps ) )
Distinct variable groups:    y, A    x, y    ph, y
Allowed substitution hints:    ph( x)    ps( x, y)    A( x)

Proof of Theorem rr19.28v
StepHypRef Expression
1 simpl 457 . . . . . 6  |-  ( (
ph  /\  ps )  ->  ph )
21ralimi 2809 . . . . 5  |-  ( A. y  e.  A  ( ph  /\  ps )  ->  A. y  e.  A  ph )
3 biidd 237 . . . . . 6  |-  ( y  =  x  ->  ( ph 
<-> 
ph ) )
43rspcv 3162 . . . . 5  |-  ( x  e.  A  ->  ( A. y  e.  A  ph 
->  ph ) )
52, 4syl5 32 . . . 4  |-  ( x  e.  A  ->  ( A. y  e.  A  ( ph  /\  ps )  ->  ph ) )
6 simpr 461 . . . . . 6  |-  ( (
ph  /\  ps )  ->  ps )
76ralimi 2809 . . . . 5  |-  ( A. y  e.  A  ( ph  /\  ps )  ->  A. y  e.  A  ps )
87a1i 11 . . . 4  |-  ( x  e.  A  ->  ( A. y  e.  A  ( ph  /\  ps )  ->  A. y  e.  A  ps ) )
95, 8jcad 533 . . 3  |-  ( x  e.  A  ->  ( A. y  e.  A  ( ph  /\  ps )  ->  ( ph  /\  A. y  e.  A  ps ) ) )
109ralimia 2807 . 2  |-  ( A. x  e.  A  A. y  e.  A  ( ph  /\  ps )  ->  A. x  e.  A  ( ph  /\  A. y  e.  A  ps )
)
11 r19.28av 2949 . . 3  |-  ( (
ph  /\  A. y  e.  A  ps )  ->  A. y  e.  A  ( ph  /\  ps )
)
1211ralimi 2809 . 2  |-  ( A. x  e.  A  ( ph  /\  A. y  e.  A  ps )  ->  A. x  e.  A  A. y  e.  A  ( ph  /\  ps )
)
1310, 12impbii 188 1  |-  ( A. x  e.  A  A. y  e.  A  ( ph  /\  ps )  <->  A. x  e.  A  ( ph  /\ 
A. y  e.  A  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1758   A.wral 2793
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 1952  ax-ext 2430
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 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ral 2798  df-v 3067
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator