MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  r19.40 Unicode version

Theorem r19.40 2653
Description: Restricted quantifier version of Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 2-Apr-2004.)
Assertion
Ref Expression
r19.40  |-  ( E. x  e.  A  (
ph  /\  ps )  ->  ( E. x  e.  A  ph  /\  E. x  e.  A  ps ) )

Proof of Theorem r19.40
StepHypRef Expression
1 simpl 445 . . 3  |-  ( (
ph  /\  ps )  ->  ph )
21reximi 2612 . 2  |-  ( E. x  e.  A  (
ph  /\  ps )  ->  E. x  e.  A  ph )
3 simpr 449 . . 3  |-  ( (
ph  /\  ps )  ->  ps )
43reximi 2612 . 2  |-  ( E. x  e.  A  (
ph  /\  ps )  ->  E. x  e.  A  ps )
52, 4jca 520 1  |-  ( E. x  e.  A  (
ph  /\  ps )  ->  ( E. x  e.  A  ph  /\  E. x  e.  A  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360   E.wrex 2510
This theorem is referenced by:  rexanuz  11706  txflf  17533  metequiv2  17888  axfelem22  23535  mzpcompact2lem  25995
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-gen 1536
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1538  df-ral 2513  df-rex 2514
  Copyright terms: Public domain W3C validator