MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  morimv Structured version   Unicode version

Theorem morimv 2331
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 6 . . . . . . 7  |-  ( ps 
->  ( ph  ->  ps ) )
21a1i 11 . . . . . 6  |-  ( ph  ->  ( ps  ->  ( ph  ->  ps ) ) )
32imim1d 72 . . . . 5  |-  ( ph  ->  ( ( ( ph  ->  ps )  ->  x  =  y )  -> 
( ps  ->  x  =  y ) ) )
43alimdv 1632 . . . 4  |-  ( ph  ->  ( A. x ( ( ph  ->  ps )  ->  x  =  y )  ->  A. x
( ps  ->  x  =  y ) ) )
54eximdv 1633 . . 3  |-  ( ph  ->  ( E. y A. x ( ( ph  ->  ps )  ->  x  =  y )  ->  E. y A. x ( ps  ->  x  =  y ) ) )
6 nfv 1630 . . . 4  |-  F/ y ( ph  ->  ps )
76mo2 2312 . . 3  |-  ( E* x ( ph  ->  ps )  <->  E. y A. x
( ( ph  ->  ps )  ->  x  =  y ) )
8 nfv 1630 . . . 4  |-  F/ y ps
98mo2 2312 . . 3  |-  ( E* x ps  <->  E. y A. x ( ps  ->  x  =  y ) )
105, 7, 93imtr4g 263 . 2  |-  ( ph  ->  ( E* x (
ph  ->  ps )  ->  E* x ps ) )
1110com12 30 1  |-  ( E* x ( ph  ->  ps )  ->  ( ph  ->  E* x ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1550   E.wex 1551   E*wmo 2284
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 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951
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 1660  df-eu 2287  df-mo 2288
  Copyright terms: Public domain W3C validator