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

Theorem elimf 5582
Description: Eliminate a mapping hypothesis for the weak deduction theorem dedth 3772, when a special case  G : A --> B is provable, in order to convert  F : A --> B from a hypothesis to an antecedent. (Contributed by NM, 24-Aug-2006.)
Hypothesis
Ref Expression
elimf.1  |-  G : A
--> B
Assertion
Ref Expression
elimf  |-  if ( F : A --> B ,  F ,  G ) : A --> B

Proof of Theorem elimf
StepHypRef Expression
1 feq1 5568 . 2  |-  ( F  =  if ( F : A --> B ,  F ,  G )  ->  ( F : A --> B 
<->  if ( F : A
--> B ,  F ,  G ) : A --> B ) )
2 feq1 5568 . 2  |-  ( G  =  if ( F : A --> B ,  F ,  G )  ->  ( G : A --> B 
<->  if ( F : A
--> B ,  F ,  G ) : A --> B ) )
3 elimf.1 . 2  |-  G : A
--> B
41, 2, 3elimhyp 3779 1  |-  if ( F : A --> B ,  F ,  G ) : A --> B
Colors of variables: wff set class
Syntax hints:   ifcif 3731   -->wf 5442
This theorem is referenced by:  hosubcl  23268  hoaddcom  23269  hoaddass  23277  hocsubdir  23280  hoaddid1  23286  hodid  23287  ho0sub  23292  honegsub  23294  hoddi  23485
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-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-fun 5448  df-fn 5449  df-f 5450
  Copyright terms: Public domain W3C validator