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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 4089, 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 5939 | . 2 ⊢ (𝐹 = if(𝐹:𝐴⟶𝐵, 𝐹, 𝐺) → (𝐹:𝐴⟶𝐵 ↔ if(𝐹:𝐴⟶𝐵, 𝐹, 𝐺):𝐴⟶𝐵)) | |
2 | feq1 5939 | . 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 4096 | 1 ⊢ if(𝐹:𝐴⟶𝐵, 𝐹, 𝐺):𝐴⟶𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ifcif 4036 ⟶wf 5800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-fun 5806 df-fn 5807 df-f 5808 |
This theorem is referenced by: hosubcl 28016 hoaddcom 28017 hoaddass 28025 hocsubdir 28028 hoaddid1 28034 hodid 28035 ho0sub 28040 honegsub 28042 hoddi 28233 |
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