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Theorem 19.12 1734
Description: Theorem 19.12 of [Margaris] p. 89. Assuming the converse is a mistake sometimes made by beginners! But sometimes the converse does hold, as in 19.12vv 1839 and r19.12sn 3696. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
19.12  |-  ( E. x A. y ph  ->  A. y E. x ph )

Proof of Theorem 19.12
StepHypRef Expression
1 hba1 1719 . . 3  |-  ( A. y ph  ->  A. y A. y ph )
21hbex 1733 . 2  |-  ( E. x A. y ph  ->  A. y E. x A. y ph )
3 sp 1716 . . 3  |-  ( A. y ph  ->  ph )
43eximi 1563 . 2  |-  ( E. x A. y ph  ->  E. x ph )
52, 4alrimih 1552 1  |-  ( E. x A. y ph  ->  A. y E. x ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1527   E.wex 1528
This theorem is referenced by:  ax12olem2  1869  pm11.61  27592
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-6 1703  ax-7 1708  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-ex 1529
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