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Theorem 19.41rg 28615
Description: Closed form of right-to-left implication of 19.41 1827, Theorem 19.41 of [Margaris] p. 90. Derived from 19.41rgVD 28994. (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 sp 1728 . . . 4  |-  ( A. x ( ps  ->  A. x ps )  -> 
( ps  ->  A. x ps ) )
2 pm3.21 435 . . . . . . 7  |-  ( ps 
->  ( ph  ->  ( ph  /\  ps ) ) )
32a1i 10 . . . . . 6  |-  ( ( ps  ->  A. x ps )  ->  ( ps 
->  ( ph  ->  ( ph  /\  ps ) ) ) )
43al2imi 1551 . . . . 5  |-  ( A. x ( ps  ->  A. x ps )  -> 
( A. x ps 
->  A. x ( ph  ->  ( ph  /\  ps ) ) ) )
5 exim 1565 . . . . 5  |-  ( A. x ( ph  ->  (
ph  /\  ps )
)  ->  ( E. x ph  ->  E. x
( ph  /\  ps )
) )
64, 5syl6 29 . . . 4  |-  ( A. x ( ps  ->  A. x ps )  -> 
( A. x ps 
->  ( E. x ph  ->  E. x ( ph  /\ 
ps ) ) ) )
71, 6syld 40 . . 3  |-  ( A. x ( ps  ->  A. x ps )  -> 
( ps  ->  ( E. x ph  ->  E. x
( ph  /\  ps )
) ) )
87com23 72 . 2  |-  ( A. x ( ps  ->  A. x ps )  -> 
( E. x ph  ->  ( ps  ->  E. x
( ph  /\  ps )
) ) )
98imp3a 420 1  |-  ( A. x ( ps  ->  A. x ps )  -> 
( ( E. x ph  /\  ps )  ->  E. x ( ph  /\  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1530   E.wex 1531
This theorem is referenced by:  a9e2nd  28623  a9e2ndVD  29000  a9e2ndALT  29023
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532
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