Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  19.41rg Unicode version

Theorem 19.41rg 27009
Description: Closed form of right-to-left implication of 19.41 1799, Theorem 19.41 of [Margaris] p. 90. Derived from 19.41rgVD 27368. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
19.41rg  |-  ( A. x ( ps  ->  A. x ps )  -> 
( ( E. x ph  /\  ps )  ->  E. x ( ph  /\  ps ) ) )

Proof of Theorem 19.41rg
StepHypRef Expression
1 ax-4 1692 . . . 4  |-  ( A. x ( ps  ->  A. x ps )  -> 
( ps  ->  A. x ps ) )
2 pm3.21 437 . . . . . . 7  |-  ( ps 
->  ( ph  ->  ( ph  /\  ps ) ) )
32a1i 12 . . . . . 6  |-  ( ( ps  ->  A. x ps )  ->  ( ps 
->  ( ph  ->  ( ph  /\  ps ) ) ) )
43al2imi 1549 . . . . 5  |-  ( A. x ( ps  ->  A. x ps )  -> 
( A. x ps 
->  A. x ( ph  ->  ( ph  /\  ps ) ) ) )
5 exim 1573 . . . . 5  |-  ( A. x ( ph  ->  (
ph  /\  ps )
)  ->  ( E. x ph  ->  E. x
( ph  /\  ps )
) )
64, 5syl6 31 . . . 4  |-  ( A. x ( ps  ->  A. x ps )  -> 
( A. x ps 
->  ( E. x ph  ->  E. x ( ph  /\ 
ps ) ) ) )
71, 6syld 42 . . 3  |-  ( A. x ( ps  ->  A. x ps )  -> 
( ps  ->  ( E. x ph  ->  E. x
( ph  /\  ps )
) ) )
87com23 74 . 2  |-  ( A. x ( ps  ->  A. x ps )  -> 
( E. x ph  ->  ( ps  ->  E. x
( ph  /\  ps )
) ) )
98imp3a 422 1  |-  ( A. x ( ps  ->  A. x ps )  -> 
( ( E. x ph  /\  ps )  ->  E. x ( ph  /\  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360   A.wal 1532   E.wex 1537
This theorem is referenced by:  a9e2nd  27017  a9e2ndVD  27374  a9e2ndALT  27397
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-gen 1536  ax-4 1692
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1538
  Copyright terms: Public domain W3C validator