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Theorem 2moswap 2218
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 1706 . . . 4  |-  F/ y E. y ph
21moexex 2212 . . 3  |-  ( ( E* x E. y ph  /\  A. x E* y ph )  ->  E* y E. x ( E. y ph  /\  ph ) )
32expcom 424 . 2  |-  ( A. x E* y ph  ->  ( E* x E. y ph  ->  E* y E. x ( E. y ph  /\  ph ) ) )
4 19.8a 1718 . . . . 5  |-  ( ph  ->  E. y ph )
54pm4.71ri 614 . . . 4  |-  ( ph  <->  ( E. y ph  /\  ph ) )
65exbii 1569 . . 3  |-  ( E. x ph  <->  E. x
( E. y ph  /\ 
ph ) )
76mobii 2179 . 2  |-  ( E* y E. x ph  <->  E* y E. x ( E. y ph  /\  ph ) )
83, 7syl6ibr 218 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 358   A.wal 1527   E.wex 1528   E*wmo 2144
This theorem is referenced by:  2euswap  2219  2eu1  2223  2rmoswap  27962
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148
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