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Theorem 2moswap 2378
Description: A condition allowing swap of "at most one" and existential quantifiers. (Contributed by NM, 10-Apr-2004.)
Assertion
Ref Expression
2moswap  |-  ( A. x E* y ph  ->  ( E* x E. y ph  ->  E* y E. x ph ) )

Proof of Theorem 2moswap
StepHypRef Expression
1 nfe1 1789 . . . 4  |-  F/ y E. y ph
21moexex 2371 . . 3  |-  ( ( E* x E. y ph  /\  A. x E* y ph )  ->  E* y E. x ( E. y ph  /\  ph ) )
32expcom 435 . 2  |-  ( A. x E* y ph  ->  ( E* x E. y ph  ->  E* y E. x ( E. y ph  /\  ph ) ) )
4 19.8a 1806 . . . . 5  |-  ( ph  ->  E. y ph )
54pm4.71ri 633 . . . 4  |-  ( ph  <->  ( E. y ph  /\  ph ) )
65exbii 1644 . . 3  |-  ( E. x ph  <->  E. x
( E. y ph  /\ 
ph ) )
76mobii 2301 . 2  |-  ( E* y E. x ph  <->  E* y E. x ( E. y ph  /\  ph ) )
83, 7syl6ibr 227 1  |-  ( A. x E* y ph  ->  ( E* x E. y ph  ->  E* y E. x ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1377   E.wex 1596   E*wmo 2276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-eu 2279  df-mo 2280
This theorem is referenced by:  2euswap  2379  2eu1OLD  2387  2rmoswap  31979
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