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Mirrors > Home > MPE Home > Th. List > 3anbi1i | Structured version Visualization version GIF version |
Description: Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.) |
Ref | Expression |
---|---|
3anbi1i.1 | ⊢ (𝜑 ↔ 𝜓) |
Ref | Expression |
---|---|
3anbi1i | ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) ↔ (𝜓 ∧ 𝜒 ∧ 𝜃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3anbi1i.1 | . 2 ⊢ (𝜑 ↔ 𝜓) | |
2 | biid 264 | . 2 ⊢ (𝜒 ↔ 𝜒) | |
3 | biid 264 | . 2 ⊢ (𝜃 ↔ 𝜃) | |
4 | 1, 2, 3 | 3anbi123i 1152 | 1 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) ↔ (𝜓 ∧ 𝜒 ∧ 𝜃)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ w3a 1084 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 210 df-an 400 df-3an 1086 |
This theorem is referenced by: iinfi 8865 fzolb 13039 brfi1uzind 13852 opfi1uzind 13855 sqrlem5 14598 bitsmod 15775 isfunc 17126 txcn 22231 trfil2 22492 isclmp 23702 eulerpartlemn 31749 bnj976 32159 bnj543 32275 bnj594 32294 bnj917 32316 topdifinffinlem 34764 dath 37032 ichexmpl1 43986 elfzolborelfzop1 44928 nnolog2flm1 45004 |
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