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

Theorem exintr 1625
Description: Introduce a conjunct in the scope of an existential quantifier. (Contributed by NM, 11-Aug-1993.)
Assertion
Ref Expression
exintr  |-  ( A. x ( ph  ->  ps )  ->  ( E. x ph  ->  E. x
( ph  /\  ps )
) )

Proof of Theorem exintr
StepHypRef Expression
1 exintrbi 1624 . 2  |-  ( A. x ( ph  ->  ps )  ->  ( E. x ph  <->  E. x ( ph  /\ 
ps ) ) )
21biimpd 200 1  |-  ( A. x ( ph  ->  ps )  ->  ( E. x ph  ->  E. x
( ph  /\  ps )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360   A.wal 1550   E.wex 1551
This theorem is referenced by:  equs4  1998  ceqsex  2991  r19.2z  3718  pwpw0  3947  pwsnALT  4011  ceqsex3OLD  26710  pm10.55  27542  bnj1023  29152  bnj1109  29158
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552
  Copyright terms: Public domain W3C validator