| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > jaao | Structured version Visualization version 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 480 | . 2 ⊢ ((𝜑 ∧ 𝜃) → (𝜓 → 𝜒)) |
| 3 | jaao.2 | . . 3 ⊢ (𝜃 → (𝜏 → 𝜒)) | |
| 4 | 3 | adantl 481 | . 2 ⊢ ((𝜑 ∧ 𝜃) → (𝜏 → 𝜒)) |
| 5 | 2, 4 | jaod 860 | 1 ⊢ ((𝜑 ∧ 𝜃) → ((𝜓 ∨ 𝜏) → 𝜒)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 |
| This theorem is referenced by: pm3.44 962 pm3.48 966 prlem1 1055 elpr2g 4651 ordtri1 6417 ordun 6488 suc11 6491 funun 6612 poxp 8153 suc11reg 9659 rankunb 9890 gruun 10846 ofpreima2 32676 wl-orel12 37512 clsk1indlem3 44056 |
| Copyright terms: Public domain | W3C validator |