MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ex-natded9.26-2 Unicode version

Theorem ex-natded9.26-2 20807
Description: A more efficient proof of Theorem 9.26 of [Clemente] p. 45. Compare with ex-natded9.26 20806. (Contributed by Mario Carneiro, 9-Feb-2017.)
Hypothesis
Ref Expression
ex-natded9.26.1  |-  ( ph  ->  E. x A. y ps )
Assertion
Ref Expression
ex-natded9.26-2  |-  ( ph  ->  A. y E. x ps )
Distinct variable group:    x, y,
ph
Allowed substitution hints:    ps( x, y)

Proof of Theorem ex-natded9.26-2
StepHypRef Expression
1 ex-natded9.26.1 . . 3  |-  ( ph  ->  E. x A. y ps )
2 sp 1716 . . . 4  |-  ( A. y ps  ->  ps )
32eximi 1563 . . 3  |-  ( E. x A. y ps 
->  E. x ps )
41, 3syl 15 . 2  |-  ( ph  ->  E. x ps )
54alrimiv 1617 1  |-  ( ph  ->  A. y E. x ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1527   E.wex 1528
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-ex 1529
  Copyright terms: Public domain W3C validator