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Mirrors > Home > ILE Home > Th. List > bibi12d | GIF version |
Description: Deduction joining two equivalences to form equivalence of biconditionals. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
imbi12d.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
imbi12d.2 | ⊢ (𝜑 → (𝜃 ↔ 𝜏)) |
Ref | Expression |
---|---|
bibi12d | ⊢ (𝜑 → ((𝜓 ↔ 𝜃) ↔ (𝜒 ↔ 𝜏))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imbi12d.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | 1 | bibi1d 233 | . 2 ⊢ (𝜑 → ((𝜓 ↔ 𝜃) ↔ (𝜒 ↔ 𝜃))) |
3 | imbi12d.2 | . . 3 ⊢ (𝜑 → (𝜃 ↔ 𝜏)) | |
4 | 3 | bibi2d 232 | . 2 ⊢ (𝜑 → ((𝜒 ↔ 𝜃) ↔ (𝜒 ↔ 𝜏))) |
5 | 2, 4 | bitrd 188 | 1 ⊢ (𝜑 → ((𝜓 ↔ 𝜃) ↔ (𝜒 ↔ 𝜏))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: pm5.32 453 bi2bian9 608 cleqh 2277 abbi 2291 cleqf 2344 cbvreuvw 2711 vtoclb 2796 vtoclbg 2800 ceqsexg 2867 elabgf 2881 reu6 2928 ru 2963 sbcbig 3011 sbcne12g 3077 sbcnestgf 3110 preq12bg 3775 nalset 4135 undifexmid 4195 exmidsssn 4204 exmidsssnc 4205 exmidundif 4208 opthg 4240 opelopabsb 4262 wetriext 4578 opeliunxp2 4769 resieq 4919 elimasng 4998 cbviota 5185 iota2df 5204 fnbrfvb 5558 fvelimab 5574 fmptco 5684 fsng 5691 fressnfv 5705 funfvima3 5752 isorel 5811 isocnv 5814 isocnv2 5815 isotr 5819 ovg 6015 caovcang 6038 caovordg 6044 caovord3d 6047 caovord 6048 opeliunxp2f 6241 dftpos4 6266 ecopovsym 6633 ecopovsymg 6636 xpf1o 6846 nneneq 6859 supmoti 6994 supsnti 7006 isotilem 7007 isoti 7008 ltanqg 7401 ltmnqg 7402 elinp 7475 prnmaxl 7489 prnminu 7490 ltasrg 7771 axpre-ltadd 7887 zextle 9346 zextlt 9347 xlesubadd 9885 rexfiuz 11000 climshft 11314 dvdsext 11863 ltoddhalfle 11900 halfleoddlt 11901 bezoutlemmo 12009 bezoutlemeu 12010 bezoutlemle 12011 bezoutlemsup 12012 dfgcd3 12013 dvdssq 12034 rpexp 12155 pcdvdsb 12321 isnsg 13067 nsgbi 13069 elnmz 13073 nmzbi 13074 nmznsg 13078 xmeteq0 13898 comet 14038 dedekindeulemuub 14134 dedekindeulemloc 14136 dedekindicclemuub 14143 dedekindicclemloc 14145 logltb 14334 bj-nalset 14686 bj-d0clsepcl 14716 bj-nn0sucALT 14769 ltlenmkv 14857 |
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