MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfim1 Structured version   Unicode version

Theorem nfim1 1830
Description: A closed form of nfim 1832. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
Hypotheses
Ref Expression
nfim1.1  |-  F/ x ph
nfim1.2  |-  ( ph  ->  F/ x ps )
Assertion
Ref Expression
nfim1  |-  F/ x
( ph  ->  ps )

Proof of Theorem nfim1
StepHypRef Expression
1 nfim1.1 . . . 4  |-  F/ x ph
21nfri 1778 . . 3  |-  ( ph  ->  A. x ph )
3 nfim1.2 . . . 4  |-  ( ph  ->  F/ x ps )
43nfrd 1779 . . 3  |-  ( ph  ->  ( ps  ->  A. x ps ) )
52, 4hbim1 1829 . 2  |-  ( (
ph  ->  ps )  ->  A. x ( ph  ->  ps ) )
65nfi 1560 1  |-  F/ x
( ph  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4   F/wnf 1553
This theorem is referenced by:  nfim  1832  cbv1  1973  dvelimdf  2066  sbied  2123  sbco2d  2162  sbco2dwAUX7  29487  sbco2dOLD7  29654
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-11 1761
This theorem depends on definitions:  df-bi 178  df-ex 1551  df-nf 1554
  Copyright terms: Public domain W3C validator