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Theorem moanim 2301
Description: Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 24-Dec-2018.)
Hypothesis
Ref Expression
moanim.1  |-  F/ x ph
Assertion
Ref Expression
moanim  |-  ( E* x ( ph  /\  ps )  <->  ( ph  ->  E* x ps ) )

Proof of Theorem moanim
StepHypRef Expression
1 moanim.1 . . . 4  |-  F/ x ph
2 ibar 502 . . . 4  |-  ( ph  ->  ( ps  <->  ( ph  /\ 
ps ) ) )
31, 2mobid 2259 . . 3  |-  ( ph  ->  ( E* x ps  <->  E* x ( ph  /\  ps ) ) )
43biimprcd 225 . 2  |-  ( E* x ( ph  /\  ps )  ->  ( ph  ->  E* x ps )
)
5 simpl 455 . . . . . 6  |-  ( (
ph  /\  ps )  ->  ph )
61, 5exlimi 1940 . . . . 5  |-  ( E. x ( ph  /\  ps )  ->  ph )
7 exmo 2265 . . . . . 6  |-  ( E. x ( ph  /\  ps )  \/  E* x ( ph  /\  ps ) )
87ori 373 . . . . 5  |-  ( -. 
E. x ( ph  /\ 
ps )  ->  E* x ( ph  /\  ps ) )
96, 8nsyl4 142 . . . 4  |-  ( -. 
E* x ( ph  /\ 
ps )  ->  ph )
109con1i 129 . . 3  |-  ( -. 
ph  ->  E* x (
ph  /\  ps )
)
11 moan 2296 . . 3  |-  ( E* x ps  ->  E* x ( ph  /\  ps ) )
1210, 11ja 161 . 2  |-  ( (
ph  ->  E* x ps )  ->  E* x
( ph  /\  ps )
)
134, 12impbii 188 1  |-  ( E* x ( ph  /\  ps )  <->  ( ph  ->  E* x ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   E.wex 1633   F/wnf 1637   E*wmo 2239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-12 1878
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-ex 1634  df-nf 1638  df-eu 2242  df-mo 2243
This theorem is referenced by:  moanimv  2303  moanmo  2304  2eu1OLD  2328
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