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Mirrors > Home > MPE Home > Th. List > feq3d | Structured version Visualization version GIF version |
Description: Equality deduction for functions. (Contributed by AV, 1-Jan-2020.) |
Ref | Expression |
---|---|
feq2d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
feq3d | ⊢ (𝜑 → (𝐹:𝑋⟶𝐴 ↔ 𝐹:𝑋⟶𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | feq2d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | feq3 6718 | . 2 ⊢ (𝐴 = 𝐵 → (𝐹:𝑋⟶𝐴 ↔ 𝐹:𝑋⟶𝐵)) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (𝐹:𝑋⟶𝐴 ↔ 𝐹:𝑋⟶𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1536 ⟶wf 6558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1776 df-cleq 2726 df-ss 3979 df-f 6566 |
This theorem is referenced by: feq123d 6725 fsn2 7155 fsng 7156 fsnunf 7204 funcres2b 17947 funcres2 17948 funcres2c 17954 catciso 18164 hofcl 18315 yonedalem4c 18333 yonedalem3b 18335 yonedainv 18337 gsumress 18707 resmgmhm2b 18738 resmhm2b 18847 pwsdiagmhm 18856 frmdup3lem 18891 frmdup3 18892 isghmOLD 19246 frgpup3lem 19809 gsumzsubmcl 19950 dmdprd 20032 zrtermorngc 20659 zrtermoringc 20691 imadrhmcl 20814 rngqiprngghm 21326 frlmup2 21836 scmatghm 22554 scmatmhm 22555 mdetdiaglem 22619 cnpf2 23273 2ndcctbss 23478 1stcelcls 23484 uptx 23648 txcn 23649 tsmssubm 24166 cnextucn 24327 pi1addf 25093 caufval 25322 equivcau 25347 lmcau 25360 plypf1 26265 coef2 26284 ulmval 26437 uhgr0vb 29103 uhgrun 29105 uhgrstrrepe 29109 isumgrs 29127 upgrun 29149 umgrun 29151 wksfval 29641 wlkres 29702 ajfval 30837 chscllem4 31668 feq3dd 32640 rlocf1 33259 imasmhm 33361 imasghm 33362 imasrhm 33363 lindfpropd 33389 ply1degltdimlem 33649 fedgmullem1 33656 rrhf 33960 sibff 34317 sibfof 34321 orvcval4 34441 bj-finsumval0 37267 matunitlindflem2 37603 poimirlem9 37615 isbnd3 37770 prdsbnd 37779 heibor 37807 elghomlem1OLD 37871 aks6d1c6lem2 42152 aks6d1c6lem3 42153 aks6d1c6isolem2 42156 aks6d1c6lem5 42158 evlsvvval 42549 cnfsmf 46695 upwlksfval 47978 |
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