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Theorem 19.12 1735
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 1840 and r19.12sn 3697. (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 1720 . . 3  |-  ( A. y ph  ->  A. y A. y ph )
21hbex 1734 . 2  |-  ( E. x A. y ph  ->  A. y E. x A. y ph )
3 ax4 1717 . . 3  |-  ( A. y ph  ->  ph )
43eximi 1564 . 2  |-  ( E. x A. y ph  ->  E. x ph )
52, 4alrimih 1553 1  |-  ( E. x A. y ph  ->  A. y E. x ph )
Colors of variables: wff set class
Syntax hints:    -> wi 6   A.wal 1528   E.wex 1529
This theorem is referenced by:  ax12olem2  1870  pm11.61  26991
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716
This theorem depends on definitions:  df-bi 179  df-ex 1530
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