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Theorem r19.41 2482
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 387 . . . 4  |-  ( ( ( x  e.  A  /\  ph )  /\  ps ) 
<->  ( x  e.  A  /\  ( ph  /\  ps ) ) )
21exbii 1512 . . 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 1592 . . 3  |-  ( E. x ( ( x  e.  A  /\  ph )  /\  ps )  <->  ( E. x ( x  e.  A  /\  ph )  /\  ps ) )
52, 4bitr3i 179 . 2  |-  ( E. x ( x  e.  A  /\  ( ph  /\ 
ps ) )  <->  ( E. x ( x  e.  A  /\  ph )  /\  ps ) )
6 df-rex 2329 . 2  |-  ( E. x  e.  A  (
ph  /\  ps )  <->  E. x ( x  e.  A  /\  ( ph  /\ 
ps ) ) )
7 df-rex 2329 . . 3  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
87anbi1i 439 . 2  |-  ( ( E. x  e.  A  ph 
/\  ps )  <->  ( E. x ( x  e.  A  /\  ph )  /\  ps ) )
95, 6, 83bitr4i 205 1  |-  ( E. x  e.  A  (
ph  /\  ps )  <->  ( E. x  e.  A  ph 
/\  ps ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    <-> wb 102   F/wnf 1365   E.wex 1397    e. wcel 1409   E.wrex 2324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-rex 2329
This theorem is referenced by:  r19.41v  2483
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