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Mirrors > Home > ILE Home > Th. List > biimp3a | GIF version |
Description: Infer implication from a logical equivalence. Similar to biimpa 280. (Contributed by NM, 4-Sep-2005.) |
Ref | Expression |
---|---|
biimp3a.1 | ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) |
Ref | Expression |
---|---|
biimp3a | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biimp3a.1 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) | |
2 | 1 | biimpa 280 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃) |
3 | 2 | 3impa 1099 | 1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∧ w3a 885 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 |
This theorem depends on definitions: df-bi 110 df-3an 887 |
This theorem is referenced by: nnawordex 6101 nn0addge1 8228 nn0addge2 8229 nn0sub2 8314 eluzp1p1 8498 uznn0sub 8504 iocssre 8822 icossre 8823 iccssre 8824 lincmb01cmp 8871 iccf1o 8872 fzosplitprm1 9090 subfzo0 9097 |
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