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Mirrors > Home > ILE Home > Th. List > 3anbi2i | 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 |
---|---|
3anbi2i | ⊢ ((𝜒 ∧ 𝜑 ∧ 𝜃) ↔ (𝜒 ∧ 𝜓 ∧ 𝜃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biid 170 | . 2 ⊢ (𝜒 ↔ 𝜒) | |
2 | 3anbi1i.1 | . 2 ⊢ (𝜑 ↔ 𝜓) | |
3 | biid 170 | . 2 ⊢ (𝜃 ↔ 𝜃) | |
4 | 1, 2, 3 | 3anbi123i 1183 | 1 ⊢ ((𝜒 ∧ 𝜑 ∧ 𝜃) ↔ (𝜒 ∧ 𝜓 ∧ 𝜃)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∧ w3a 973 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem depends on definitions: df-bi 116 df-3an 975 |
This theorem is referenced by: seq3f1olemp 10451 seq3f1oleml 10452 fsum3 11343 |
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