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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 4400, 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 6322 | . 2 ⊢ (𝐹 = if(𝐹:𝐴⟶𝐵, 𝐹, 𝐺) → (𝐹:𝐴⟶𝐵 ↔ if(𝐹:𝐴⟶𝐵, 𝐹, 𝐺):𝐴⟶𝐵)) | |
2 | feq1 6322 | . 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 4407 | 1 ⊢ if(𝐹:𝐴⟶𝐵, 𝐹, 𝐺):𝐴⟶𝐵 |
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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