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Mirrors > Home > ILE Home > Th. List > biimp3ar | GIF version |
Description: Infer implication from a logical equivalence. Similar to biimpar 295. (Contributed by NM, 2-Jan-2009.) |
Ref | Expression |
---|---|
biimp3a.1 | ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) |
Ref | Expression |
---|---|
biimp3ar | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biimp3a.1 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) | |
2 | 1 | exbiri 380 | . 2 ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒))) |
3 | 2 | 3imp 1188 | 1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ 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: rmoi 3048 brelrng 4842 ssfzo12 10180 abssubge0 11066 qredeu 12051 basgen2 12875 logbprmirr 13684 lgssq 13735 lgssq2 13736 |
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