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Theorem elimf 5388
Description: Eliminate a mapping hypothesis for the weak deduction theorem dedth 3606, 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 5375 . 2  |-  ( F  =  if ( F : A --> B ,  F ,  G )  ->  ( F : A --> B 
<->  if ( F : A
--> B ,  F ,  G ) : A --> B ) )
2 feq1 5375 . 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 3613 1  |-  if ( F : A --> B ,  F ,  G ) : A --> B
Colors of variables: wff set class
Syntax hints:   ifcif 3565   -->wf 5251
This theorem is referenced by:  hosubcl  22353  hoaddcom  22354  hoaddass  22362  hocsubdir  22365  hoaddid1  22371  hodid  22372  ho0sub  22377  honegsub  22379  hoddi  22570
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  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-fun 5257  df-fn 5258  df-f 5259
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