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Theorem 2moswap 2353
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 1894 . . . 4  |-  F/ y E. y ph
21moexex 2347 . . 3  |-  ( ( E* x E. y ph  /\  A. x E* y ph )  ->  E* y E. x ( E. y ph  /\  ph ) )
32expcom 436 . 2  |-  ( A. x E* y ph  ->  ( E* x E. y ph  ->  E* y E. x ( E. y ph  /\  ph ) ) )
4 19.8a 1912 . . . . 5  |-  ( ph  ->  E. y ph )
54pm4.71ri 637 . . . 4  |-  ( ph  <->  ( E. y ph  /\  ph ) )
65exbii 1712 . . 3  |-  ( E. x ph  <->  E. x
( E. y ph  /\ 
ph ) )
76mobii 2299 . 2  |-  ( E* y E. x ph  <->  E* y E. x ( E. y ph  /\  ph ) )
83, 7syl6ibr 230 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 370   A.wal 1435   E.wex 1657   E*wmo 2277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-eu 2280  df-mo 2281
This theorem is referenced by:  2euswap  2354  2rmoswap  38419
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