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Theorem 2moswap 2363
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 1750 . . . 4  |-  F/ y E. y ph
21moexex 2357 . . 3  |-  ( ( E* x E. y ph  /\  A. x E* y ph )  ->  E* y E. x ( E. y ph  /\  ph ) )
32expcom 426 . 2  |-  ( A. x E* y ph  ->  ( E* x E. y ph  ->  E* y E. x ( E. y ph  /\  ph ) ) )
4 19.8a 1765 . . . . 5  |-  ( ph  ->  E. y ph )
54pm4.71ri 616 . . . 4  |-  ( ph  <->  ( E. y ph  /\  ph ) )
65exbii 1593 . . 3  |-  ( E. x ph  <->  E. x
( E. y ph  /\ 
ph ) )
76mobii 2324 . 2  |-  ( E* y E. x ph  <->  E* y E. x ( E. y ph  /\  ph ) )
83, 7syl6ibr 220 1  |-  ( A. x E* y ph  ->  ( E* x E. y ph  ->  E* y E. x ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360   A.wal 1550   E.wex 1551   E*wmo 2289
This theorem is referenced by:  2euswap  2364  2eu1  2368  2rmoswap  28050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293
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