Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > a2and | Structured version Visualization version GIF version |
Description: Deduction distributing a conjunction as embedded antecedent. (Contributed by AV, 25-Oct-2019.) (Proof shortened by Wolf Lammen, 19-Jan-2020.) |
Ref | Expression |
---|---|
a2and.1 | ⊢ (𝜑 → ((𝜓 ∧ 𝜌) → (𝜏 → 𝜃))) |
a2and.2 | ⊢ (𝜑 → ((𝜓 ∧ 𝜌) → 𝜒)) |
Ref | Expression |
---|---|
a2and | ⊢ (𝜑 → (((𝜓 ∧ 𝜒) → 𝜏) → ((𝜓 ∧ 𝜌) → 𝜃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | a2and.2 | . . . . . . 7 ⊢ (𝜑 → ((𝜓 ∧ 𝜌) → 𝜒)) | |
2 | 1 | expd 452 | . . . . . 6 ⊢ (𝜑 → (𝜓 → (𝜌 → 𝜒))) |
3 | 2 | imdistand 728 | . . . . 5 ⊢ (𝜑 → ((𝜓 ∧ 𝜌) → (𝜓 ∧ 𝜒))) |
4 | 3 | imp 445 | . . . 4 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜌)) → (𝜓 ∧ 𝜒)) |
5 | a2and.1 | . . . . 5 ⊢ (𝜑 → ((𝜓 ∧ 𝜌) → (𝜏 → 𝜃))) | |
6 | 5 | imp 445 | . . . 4 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜌)) → (𝜏 → 𝜃)) |
7 | 4, 6 | embantd 59 | . . 3 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜌)) → (((𝜓 ∧ 𝜒) → 𝜏) → 𝜃)) |
8 | 7 | ex 450 | . 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜌) → (((𝜓 ∧ 𝜒) → 𝜏) → 𝜃))) |
9 | 8 | com23 86 | 1 ⊢ (𝜑 → (((𝜓 ∧ 𝜒) → 𝜏) → ((𝜓 ∧ 𝜌) → 𝜃))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ 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-an 386 |
This theorem is referenced by: telgsumfzs 18386 |
Copyright terms: Public domain | W3C validator |