Users' Mathboxes Mathbox for Andrew Salmon < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pm11.61 Structured version   Unicode version

Theorem pm11.61 27560
Description: Theorem *11.61 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
Assertion
Ref Expression
pm11.61  |-  ( E. y A. x (
ph  ->  ps )  ->  A. x ( ph  ->  E. y ps ) )
Distinct variable group:    ph, y
Allowed substitution hints:    ph( x)    ps( x, y)

Proof of Theorem pm11.61
StepHypRef Expression
1 19.12 1869 . 2  |-  ( E. y A. x (
ph  ->  ps )  ->  A. x E. y (
ph  ->  ps ) )
2 19.37v 1922 . . . 4  |-  ( E. y ( ph  ->  ps )  <->  ( ph  ->  E. y ps ) )
32biimpi 187 . . 3  |-  ( E. y ( ph  ->  ps )  ->  ( ph  ->  E. y ps )
)
43alimi 1568 . 2  |-  ( A. x E. y ( ph  ->  ps )  ->  A. x
( ph  ->  E. y ps ) )
51, 4syl 16 1  |-  ( E. y A. x (
ph  ->  ps )  ->  A. x ( ph  ->  E. y ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1549   E.wex 1550
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-7 1749  ax-11 1761
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554
  Copyright terms: Public domain W3C validator