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Mirrors > Home > ILE Home > Th. List > jaoa | GIF version |
Description: Inference disjoining and conjoining the antecedents of two implications. (Contributed by Stefan Allan, 1-Nov-2008.) |
Ref | Expression |
---|---|
jaao.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
jaao.2 | ⊢ (𝜃 → (𝜏 → 𝜒)) |
Ref | Expression |
---|---|
jaoa | ⊢ ((𝜑 ∨ 𝜃) → ((𝜓 ∧ 𝜏) → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | jaao.1 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
2 | 1 | adantrd 264 | . 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜏) → 𝜒)) |
3 | jaao.2 | . . 3 ⊢ (𝜃 → (𝜏 → 𝜒)) | |
4 | 3 | adantld 263 | . 2 ⊢ (𝜃 → ((𝜓 ∧ 𝜏) → 𝜒)) |
5 | 2, 4 | jaoi 636 | 1 ⊢ ((𝜑 ∨ 𝜃) → ((𝜓 ∧ 𝜏) → 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∨ wo 629 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: pm4.79dc 809 |
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