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

Theorem elimf 6182
Description: Eliminate a mapping hypothesis for the weak deduction theorem dedth 4278, when a special case 𝐺:𝐴𝐵 is provable, in order to convert 𝐹:𝐴𝐵 from a hypothesis to an antecedent. (Contributed by NM, 24-Aug-2006.)
Hypothesis
Ref Expression
elimf.1 𝐺:𝐴𝐵
Assertion
Ref Expression
elimf if(𝐹:𝐴𝐵, 𝐹, 𝐺):𝐴𝐵

Proof of Theorem elimf
StepHypRef Expression
1 feq1 6164 . 2 (𝐹 = if(𝐹:𝐴𝐵, 𝐹, 𝐺) → (𝐹:𝐴𝐵 ↔ if(𝐹:𝐴𝐵, 𝐹, 𝐺):𝐴𝐵))
2 feq1 6164 . 2 (𝐺 = if(𝐹:𝐴𝐵, 𝐹, 𝐺) → (𝐺:𝐴𝐵 ↔ if(𝐹:𝐴𝐵, 𝐹, 𝐺):𝐴𝐵))
3 elimf.1 . 2 𝐺:𝐴𝐵
41, 2, 3elimhyp 4285 1 if(𝐹:𝐴𝐵, 𝐹, 𝐺):𝐴𝐵
Colors of variables: wff setvar class
Syntax hints:  ifcif 4225  wf 6025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-fun 6031  df-fn 6032  df-f 6033
This theorem is referenced by:  hosubcl  28968  hoaddcom  28969  hoaddass  28977  hocsubdir  28980  hoaddid1  28986  hodid  28987  ho0sub  28992  honegsub  28994  hoddi  29185
  Copyright terms: Public domain W3C validator