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Mirrors > Home > MPE Home > Th. List > 3imp3i2an | Structured version Visualization version GIF version |
Description: An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 13-Apr-2022.) |
Ref | Expression |
---|---|
3imp3i2an.1 | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) |
3imp3i2an.2 | ⊢ ((𝜑 ∧ 𝜒) → 𝜏) |
3imp3i2an.3 | ⊢ ((𝜃 ∧ 𝜏) → 𝜂) |
Ref | Expression |
---|---|
3imp3i2an | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3imp3i2an.1 | . 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | |
2 | 3simpb 1079 | . . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜑 ∧ 𝜒)) | |
3 | 3imp3i2an.2 | . . 3 ⊢ ((𝜑 ∧ 𝜒) → 𝜏) | |
4 | 2, 3 | syl 17 | . 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜏) |
5 | 3imp3i2an.3 | . 2 ⊢ ((𝜃 ∧ 𝜏) → 𝜂) | |
6 | 1, 4, 5 | syl2anc 694 | 1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1054 |
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 385 df-3an 1056 |
This theorem is referenced by: cplgr3v 26387 upgr2pthnlp 26684 frgrreg 27381 eliuniin 39593 eliuniin2 39617 |
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