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Theorem elimf 5530
Description: Eliminate a mapping hypothesis for the weak deduction theorem dedth 3723, 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 5516 . 2  |-  ( F  =  if ( F : A --> B ,  F ,  G )  ->  ( F : A --> B 
<->  if ( F : A
--> B ,  F ,  G ) : A --> B ) )
2 feq1 5516 . 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 3730 1  |-  if ( F : A --> B ,  F ,  G ) : A --> B
Colors of variables: wff set class
Syntax hints:   ifcif 3682   -->wf 5390
This theorem is referenced by:  hosubcl  23124  hoaddcom  23125  hoaddass  23133  hocsubdir  23136  hoaddid1  23142  hodid  23143  ho0sub  23148  honegsub  23150  hoddi  23341
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-br 4154  df-opab 4208  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-fun 5396  df-fn 5397  df-f 5398
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