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Theorem rr19.28v 2878
Description: Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. We don't need the non-empty class condition of r19.28zv 3510 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 445 . . . . . 6  |-  ( (
ph  /\  ps )  ->  ph )
21ralimi 2591 . . . . 5  |-  ( A. y  e.  A  ( ph  /\  ps )  ->  A. y  e.  A  ph )
3 biidd 230 . . . . . 6  |-  ( y  =  x  ->  ( ph 
<-> 
ph ) )
43rcla4v 2848 . . . . 5  |-  ( x  e.  A  ->  ( A. y  e.  A  ph 
->  ph ) )
52, 4syl5 30 . . . 4  |-  ( x  e.  A  ->  ( A. y  e.  A  ( ph  /\  ps )  ->  ph ) )
6 simpr 449 . . . . . 6  |-  ( (
ph  /\  ps )  ->  ps )
76ralimi 2591 . . . . 5  |-  ( A. y  e.  A  ( ph  /\  ps )  ->  A. y  e.  A  ps )
87a1i 12 . . . 4  |-  ( x  e.  A  ->  ( A. y  e.  A  ( ph  /\  ps )  ->  A. y  e.  A  ps ) )
95, 8jcad 521 . . 3  |-  ( x  e.  A  ->  ( A. y  e.  A  ( ph  /\  ps )  ->  ( ph  /\  A. y  e.  A  ps ) ) )
109ralimia 2589 . 2  |-  ( A. x  e.  A  A. y  e.  A  ( ph  /\  ps )  ->  A. x  e.  A  ( ph  /\  A. y  e.  A  ps )
)
11 r19.28av 2655 . . 3  |-  ( (
ph  /\  A. y  e.  A  ps )  ->  A. y  e.  A  ( ph  /\  ps )
)
1211ralimi 2591 . 2  |-  ( A. x  e.  A  ( ph  /\  A. y  e.  A  ps )  ->  A. x  e.  A  A. y  e.  A  ( ph  /\  ps )
)
1310, 12impbii 182 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 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2516
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 1927  ax-ext 2237
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 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ral 2521  df-v 2759
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