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Theorem r19.41 2693
Description: Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 1-Nov-2010.)
Hypothesis
Ref Expression
r19.41.1  |-  F/ x ps
Assertion
Ref Expression
r19.41  |-  ( E. x  e.  A  (
ph  /\  ps )  <->  ( E. x  e.  A  ph 
/\  ps ) )

Proof of Theorem r19.41
StepHypRef Expression
1 anass 632 . . . 4  |-  ( ( ( x  e.  A  /\  ph )  /\  ps ) 
<->  ( x  e.  A  /\  ( ph  /\  ps ) ) )
21exbii 1570 . . 3  |-  ( E. x ( ( x  e.  A  /\  ph )  /\  ps )  <->  E. x
( x  e.  A  /\  ( ph  /\  ps ) ) )
3 r19.41.1 . . . 4  |-  F/ x ps
4319.41 1816 . . 3  |-  ( E. x ( ( x  e.  A  /\  ph )  /\  ps )  <->  ( E. x ( x  e.  A  /\  ph )  /\  ps ) )
52, 4bitr3i 244 . 2  |-  ( E. x ( x  e.  A  /\  ( ph  /\ 
ps ) )  <->  ( E. x ( x  e.  A  /\  ph )  /\  ps ) )
6 df-rex 2550 . 2  |-  ( E. x  e.  A  (
ph  /\  ps )  <->  E. x ( x  e.  A  /\  ( ph  /\ 
ps ) ) )
7 df-rex 2550 . . 3  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
87anbi1i 678 . 2  |-  ( ( E. x  e.  A  ph 
/\  ps )  <->  ( E. x ( x  e.  A  /\  ph )  /\  ps ) )
95, 6, 83bitr4i 270 1  |-  ( E. x  e.  A  (
ph  /\  ps )  <->  ( E. x  e.  A  ph 
/\  ps ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360   E.wex 1529   F/wnf 1532    e. wcel 1685   E.wrex 2545
This theorem is referenced by:  r19.41v  2694
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-11 1716
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1530  df-nf 1533  df-rex 2550
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