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Theorem elimf 6340
Description: Eliminate a mapping hypothesis for the weak deduction theorem dedth 4400, 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 6322 . 2 (𝐹 = if(𝐹:𝐴𝐵, 𝐹, 𝐺) → (𝐹:𝐴𝐵 ↔ if(𝐹:𝐴𝐵, 𝐹, 𝐺):𝐴𝐵))
2 feq1 6322 . 2 (𝐺 = if(𝐹:𝐴𝐵, 𝐹, 𝐺) → (𝐺:𝐴𝐵 ↔ if(𝐹:𝐴𝐵, 𝐹, 𝐺):𝐴𝐵))
3 elimf.1 . 2 𝐺:𝐴𝐵
41, 2, 3elimhyp 4407 1 if(𝐹:𝐴𝐵, 𝐹, 𝐺):𝐴𝐵
Colors of variables: wff setvar class
Syntax hints:  ifcif 4344  wf 6181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2744
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-rab 3091  df-v 3411  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-br 4926  df-opab 4988  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-fun 6187  df-fn 6188  df-f 6189
This theorem is referenced by:  hosubcl  29343  hoaddcom  29344  hoaddass  29352  hocsubdir  29355  hoaddid1  29361  hodid  29362  ho0sub  29367  honegsub  29369  hoddi  29560
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