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Mirrors > Home > ILE Home > Th. List > jaao | GIF version |
Description: Inference conjoining and disjoining the antecedents of two implications. (Contributed by NM, 30-Sep-1999.) |
Ref | Expression |
---|---|
jaao.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
jaao.2 | ⊢ (𝜃 → (𝜏 → 𝜒)) |
Ref | Expression |
---|---|
jaao | ⊢ ((𝜑 ∧ 𝜃) → ((𝜓 ∨ 𝜏) → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | jaao.1 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
2 | 1 | adantr 274 | . 2 ⊢ ((𝜑 ∧ 𝜃) → (𝜓 → 𝜒)) |
3 | jaao.2 | . . 3 ⊢ (𝜃 → (𝜏 → 𝜒)) | |
4 | 3 | adantl 275 | . 2 ⊢ ((𝜑 ∧ 𝜃) → (𝜏 → 𝜒)) |
5 | 2, 4 | jaod 707 | 1 ⊢ ((𝜑 ∧ 𝜃) → ((𝜓 ∨ 𝜏) → 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∨ wo 698 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: pm3.48 775 prlem1 962 nford 1554 funun 5229 poxp 6194 nntri3or 6455 |
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