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Mirrors > Home > ILE Home > Th. List > bitr2id | GIF version |
Description: A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
bitr2id.1 | ⊢ (𝜑 ↔ 𝜓) |
bitr2id.2 | ⊢ (𝜒 → (𝜓 ↔ 𝜃)) |
Ref | Expression |
---|---|
bitr2id | ⊢ (𝜒 → (𝜃 ↔ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bitr2id.1 | . . 3 ⊢ (𝜑 ↔ 𝜓) | |
2 | bitr2id.2 | . . 3 ⊢ (𝜒 → (𝜓 ↔ 𝜃)) | |
3 | 1, 2 | bitrid 192 | . 2 ⊢ (𝜒 → (𝜑 ↔ 𝜃)) |
4 | 3 | bicomd 141 | 1 ⊢ (𝜒 → (𝜃 ↔ 𝜑)) |
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: bitr3di 195 pm5.17dc 904 dn1dc 960 csbabg 3119 uniiunlem 3245 inimasn 5047 cnvpom 5172 fnresdisj 5327 f1oiso 5827 reldm 6187 mptelixpg 6734 1idprl 7589 1idpru 7590 nndiv 8960 fzn 10042 fz1sbc 10096 grpid 12912 metrest 14009 |
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