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

Theorem r19.40 2892
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 457 . . 3  |-  ( (
ph  /\  ps )  ->  ph )
21reximi 2844 . 2  |-  ( E. x  e.  A  (
ph  /\  ps )  ->  E. x  e.  A  ph )
3 simpr 461 . . 3  |-  ( (
ph  /\  ps )  ->  ps )
43reximi 2844 . 2  |-  ( E. x  e.  A  (
ph  /\  ps )  ->  E. x  e.  A  ps )
52, 4jca 532 1  |-  ( E. x  e.  A  (
ph  /\  ps )  ->  ( E. x  e.  A  ph  /\  E. x  e.  A  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   E.wrex 2737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1587  df-ral 2741  df-rex 2742
This theorem is referenced by:  rexanuz  12854  txflf  19601  metequiv2  20107  mzpcompact2lem  29114
  Copyright terms: Public domain W3C validator