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Theorem elimf 6668
Description: Eliminate a mapping hypothesis for the weak deduction theorem dedth 4545, 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 6650 . 2 (𝐹 = if(𝐹:𝐴𝐵, 𝐹, 𝐺) → (𝐹:𝐴𝐵 ↔ if(𝐹:𝐴𝐵, 𝐹, 𝐺):𝐴𝐵))
2 feq1 6650 . 2 (𝐺 = if(𝐹:𝐴𝐵, 𝐹, 𝐺) → (𝐺:𝐴𝐵 ↔ if(𝐹:𝐴𝐵, 𝐹, 𝐺):𝐴𝐵))
3 elimf.1 . 2 𝐺:𝐴𝐵
41, 2, 3elimhyp 4552 1 if(𝐹:𝐴𝐵, 𝐹, 𝐺):𝐴𝐵
Colors of variables: wff setvar class
Syntax hints:  ifcif 4487  wf 6493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-fun 6499  df-fn 6500  df-f 6501
This theorem is referenced by:  hosubcl  30757  hoaddcom  30758  hoaddass  30766  hocsubdir  30769  hoaddid1  30775  hodid  30776  ho0sub  30781  honegsub  30783  hoddi  30974
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