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Mirrors > Home > MPE Home > Th. List > elimf | Structured version Visualization version GIF version |
Description: Eliminate a mapping hypothesis for the weak deduction theorem dedth 4589, 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.) |
Ref | Expression |
---|---|
elimf.1 | ⊢ 𝐺:𝐴⟶𝐵 |
Ref | Expression |
---|---|
elimf | ⊢ if(𝐹:𝐴⟶𝐵, 𝐹, 𝐺):𝐴⟶𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | feq1 6717 | . 2 ⊢ (𝐹 = if(𝐹:𝐴⟶𝐵, 𝐹, 𝐺) → (𝐹:𝐴⟶𝐵 ↔ if(𝐹:𝐴⟶𝐵, 𝐹, 𝐺):𝐴⟶𝐵)) | |
2 | feq1 6717 | . 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 | |
4 | 1, 2, 3 | elimhyp 4596 | 1 ⊢ if(𝐹:𝐴⟶𝐵, 𝐹, 𝐺):𝐴⟶𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ifcif 4531 ⟶wf 6559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-fun 6565 df-fn 6566 df-f 6567 |
This theorem is referenced by: hosubcl 31802 hoaddcom 31803 hoaddass 31811 hocsubdir 31814 hoaddrid 31820 hodid 31821 ho0sub 31826 honegsub 31828 hoddi 32019 |
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