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Theorem rr19.28v 2910
Description: Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. We don't need the non-empty class condition of r19.28zv 3549 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 443 . . . . . 6  |-  ( (
ph  /\  ps )  ->  ph )
21ralimi 2618 . . . . 5  |-  ( A. y  e.  A  ( ph  /\  ps )  ->  A. y  e.  A  ph )
3 biidd 228 . . . . . 6  |-  ( y  =  x  ->  ( ph 
<-> 
ph ) )
43rspcv 2880 . . . . 5  |-  ( x  e.  A  ->  ( A. y  e.  A  ph 
->  ph ) )
52, 4syl5 28 . . . 4  |-  ( x  e.  A  ->  ( A. y  e.  A  ( ph  /\  ps )  ->  ph ) )
6 simpr 447 . . . . . 6  |-  ( (
ph  /\  ps )  ->  ps )
76ralimi 2618 . . . . 5  |-  ( A. y  e.  A  ( ph  /\  ps )  ->  A. y  e.  A  ps )
87a1i 10 . . . 4  |-  ( x  e.  A  ->  ( A. y  e.  A  ( ph  /\  ps )  ->  A. y  e.  A  ps ) )
95, 8jcad 519 . . 3  |-  ( x  e.  A  ->  ( A. y  e.  A  ( ph  /\  ps )  ->  ( ph  /\  A. y  e.  A  ps ) ) )
109ralimia 2616 . 2  |-  ( A. x  e.  A  A. y  e.  A  ( ph  /\  ps )  ->  A. x  e.  A  ( ph  /\  A. y  e.  A  ps )
)
11 r19.28av 2682 . . 3  |-  ( (
ph  /\  A. y  e.  A  ps )  ->  A. y  e.  A  ( ph  /\  ps )
)
1211ralimi 2618 . 2  |-  ( A. x  e.  A  ( ph  /\  A. y  e.  A  ps )  ->  A. x  e.  A  A. y  e.  A  ( ph  /\  ps )
)
1310, 12impbii 180 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-v 2790
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