MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  r19.41 Unicode version

Theorem r19.41 2852
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 631 . . . 4  |-  ( ( ( x  e.  A  /\  ph )  /\  ps ) 
<->  ( x  e.  A  /\  ( ph  /\  ps ) ) )
21exbii 1592 . . 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 1900 . . 3  |-  ( E. x ( ( x  e.  A  /\  ph )  /\  ps )  <->  ( E. x ( x  e.  A  /\  ph )  /\  ps ) )
52, 4bitr3i 243 . 2  |-  ( E. x ( x  e.  A  /\  ( ph  /\ 
ps ) )  <->  ( E. x ( x  e.  A  /\  ph )  /\  ps ) )
6 df-rex 2703 . 2  |-  ( E. x  e.  A  (
ph  /\  ps )  <->  E. x ( x  e.  A  /\  ( ph  /\ 
ps ) ) )
7 df-rex 2703 . . 3  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
87anbi1i 677 . 2  |-  ( ( E. x  e.  A  ph 
/\  ps )  <->  ( E. x ( x  e.  A  /\  ph )  /\  ps ) )
95, 6, 83bitr4i 269 1  |-  ( E. x  e.  A  (
ph  /\  ps )  <->  ( E. x  e.  A  ph 
/\  ps ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1550   F/wnf 1553    e. wcel 1725   E.wrex 2698
This theorem is referenced by:  r19.41v  2853
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-11 1761
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554  df-rex 2703
  Copyright terms: Public domain W3C validator