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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 481 | . 2 ⊢ ((𝜑 ∧ 𝜃) → (𝜓 → 𝜒)) |
3 | jaao.2 | . . 3 ⊢ (𝜃 → (𝜏 → 𝜒)) | |
4 | 3 | adantl 482 | . 2 ⊢ ((𝜑 ∧ 𝜃) → (𝜏 → 𝜒)) |
5 | 2, 4 | jaod 395 | 1 ⊢ ((𝜑 ∧ 𝜃) → ((𝜓 ∨ 𝜏) → 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 383 ∧ wa 384 |
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 197 df-or 385 df-an 386 |
This theorem is referenced by: pm3.44 533 pm3.48 878 prlem1 1005 ordtri1 5756 ordun 5829 suc11 5831 funun 5932 poxp 7289 suc11reg 8516 rankunb 8713 gruun 9628 ofpreima2 29466 wl-orel12 33294 clsk1indlem3 38341 |
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