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| Mirrors > Home > MPE Home > Th. List > animpimp2impd | Structured version Visualization version GIF version | ||
| Description: Deduction deriving nested implications from conjunctions. (Contributed by AV, 21-Aug-2022.) |
| Ref | Expression |
|---|---|
| animpimp2impd.1 | ⊢ ((𝜓 ∧ 𝜑) → (𝜒 → (𝜃 → 𝜂))) |
| animpimp2impd.2 | ⊢ ((𝜓 ∧ (𝜑 ∧ 𝜃)) → (𝜂 → 𝜏)) |
| Ref | Expression |
|---|---|
| animpimp2impd | ⊢ (𝜑 → ((𝜓 → 𝜒) → (𝜓 → (𝜃 → 𝜏)))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | animpimp2impd.1 | . . . 4 ⊢ ((𝜓 ∧ 𝜑) → (𝜒 → (𝜃 → 𝜂))) | |
| 2 | animpimp2impd.2 | . . . . . 6 ⊢ ((𝜓 ∧ (𝜑 ∧ 𝜃)) → (𝜂 → 𝜏)) | |
| 3 | 2 | expr 456 | . . . . 5 ⊢ ((𝜓 ∧ 𝜑) → (𝜃 → (𝜂 → 𝜏))) |
| 4 | 3 | a2d 29 | . . . 4 ⊢ ((𝜓 ∧ 𝜑) → ((𝜃 → 𝜂) → (𝜃 → 𝜏))) |
| 5 | 1, 4 | syld 47 | . . 3 ⊢ ((𝜓 ∧ 𝜑) → (𝜒 → (𝜃 → 𝜏))) |
| 6 | 5 | expcom 413 | . 2 ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → 𝜏)))) |
| 7 | 6 | a2d 29 | 1 ⊢ (𝜑 → ((𝜓 → 𝜒) → (𝜓 → (𝜃 → 𝜏)))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 |
| 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 |
| This theorem is referenced by: seqcl2 14061 seqfveq2 14065 seqshft2 14069 monoord 14073 seqsplit 14076 seqid2 14089 seqhomo 14090 sylow1lem1 19616 imasdsf1olem 24383 ovolicc2lem3 25554 dvnres 25967 cvmliftlem7 35296 cvmliftlem10 35299 monoordxrv 45492 smonoord 47358 |
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