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Theorem 19.12 1869
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 1921 and r19.12sn 3872. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Jan-2018.)
Assertion
Ref Expression
19.12  |-  ( E. x A. y ph  ->  A. y E. x ph )

Proof of Theorem 19.12
StepHypRef Expression
1 nfa1 1806 . . 3  |-  F/ y A. y ph
21nfex 1865 . 2  |-  F/ y E. x A. y ph
3 sp 1763 . . 3  |-  ( A. y ph  ->  ph )
43eximi 1585 . 2  |-  ( E. x A. y ph  ->  E. x ph )
52, 4alrimi 1781 1  |-  ( E. x A. y ph  ->  A. y E. x ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1549   E.wex 1550
This theorem is referenced by:  ax12olem2OLD  2012  pm11.61  27569
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-ex 1551  df-nf 1554
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