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Theorem morimv 2191
Description: Move antecedent outside of "at most one." (Contributed by NM, 28-Jul-1995.)
Assertion
Ref Expression
morimv  |-  ( E* x ( ph  ->  ps )  ->  ( ph  ->  E* x ps )
)
Distinct variable group:    ph, x
Allowed substitution hint:    ps( x)

Proof of Theorem morimv
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ax-1 5 . . . . . . 7  |-  ( ps 
->  ( ph  ->  ps ) )
21a1i 10 . . . . . 6  |-  ( ph  ->  ( ps  ->  ( ph  ->  ps ) ) )
32imim1d 69 . . . . 5  |-  ( ph  ->  ( ( ( ph  ->  ps )  ->  x  =  y )  -> 
( ps  ->  x  =  y ) ) )
43alimdv 1607 . . . 4  |-  ( ph  ->  ( A. x ( ( ph  ->  ps )  ->  x  =  y )  ->  A. x
( ps  ->  x  =  y ) ) )
54eximdv 1608 . . 3  |-  ( ph  ->  ( E. y A. x ( ( ph  ->  ps )  ->  x  =  y )  ->  E. y A. x ( ps  ->  x  =  y ) ) )
6 nfv 1605 . . . 4  |-  F/ y ( ph  ->  ps )
76mo2 2172 . . 3  |-  ( E* x ( ph  ->  ps )  <->  E. y A. x
( ( ph  ->  ps )  ->  x  =  y ) )
8 nfv 1605 . . . 4  |-  F/ y ps
98mo2 2172 . . 3  |-  ( E* x ps  <->  E. y A. x ( ps  ->  x  =  y ) )
105, 7, 93imtr4g 261 . 2  |-  ( ph  ->  ( E* x (
ph  ->  ps )  ->  E* x ps ) )
1110com12 27 1  |-  ( E* x ( ph  ->  ps )  ->  ( ph  ->  E* x ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1527   E.wex 1528    = wceq 1623   E*wmo 2144
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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