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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 4278, 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 6164 | . 2 ⊢ (𝐹 = if(𝐹:𝐴⟶𝐵, 𝐹, 𝐺) → (𝐹:𝐴⟶𝐵 ↔ if(𝐹:𝐴⟶𝐵, 𝐹, 𝐺):𝐴⟶𝐵)) | |
2 | feq1 6164 | . 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 4285 | 1 ⊢ if(𝐹:𝐴⟶𝐵, 𝐹, 𝐺):𝐴⟶𝐵 |
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 |
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